Good answers already, but consider this alternative that solves the direct problem:

Spend time with your family by videochatting with them. Maybe that would be good enough for them and you.

Setting your objective as "making them understand" is going to make progress very difficult. Tim has made a similar observation in an earlier answer, but I disagree with Tim's conclusion that you should negotiate a common position on air travel and visits home: that would be desirable, but may well be impossible to achieve.

Deep disagreements between generations are not a new thing, though the topic of disagreement may change over time. The difficulty here is that you are challenging something very fundamental about the way they live. I'm sorry if this is a bit personal, but do you want to stop your sister from flying home regularly? What would the impact on your parents be if she did?

Given that inter-generational disagreements are very difficult to deal with, a setting a more modest objective might be a good idea (this might be what you meant -- there is some ambiguity in the phrasing of the question): Can you achieve mutual acceptance of each others' positions?

This doesn't mean conceding that their views on air travel are correct -- it just means agreeing to disagree (for the time being) on that topic and working out how to maintain family relations while respecting each others views.

At the moment, it appears that your parents don't really believe that flying is the thing stopping you from vising more often, despite the fact that you have told them and gone to the length of travelling overland.

Is there something about your life-style that might confuse them? If you fly regularly for leisure, for instance, then the obvious solution would be to do less of that and fly home some -- but I'll assume it is not that simple. If you fly regularly for work, they might think that your living away is somehow related to work and the distinction between flying for work and flying to visit them is a bit arbitrary. Is your distance from home a deliberate move away,a temporary exploration, or something imposed by career? Work-related emissions bring up a lot more awkward questions.

Simplicity is really important here: it is not going to work if you discuss each flight, each tonne of carbon. It is not just about understanding the bare statements, it is about understanding the whole decision process around them. Try to go back to basics. For example, you could set yourself an emissions target, e.g. 4 tonnes CO2 equivalent per year (too much, but much better than the European average -- if you set it much lower you will have to cycle home), and ask your parents to respect this objective at the same time as respecting the fact that they don't have personal emissions targets.

How can I make my family understand my CO2-emissions-related hesitation to fly in order to spend time with them?

You want to keep your carbon footprint as low as possible, I get that. But, I feel I need to frame-challenge this question:

Right now, the message you are sending your family, is that your high morals (righteous and laudable as they may be!), are more important than your familial connection. They think it's a poor excuse, and no amount of carbon emission doomsaying is going to change that.

My advice? Rather than seeing flying as an obstacle, you should look for solutions that will allow you to meet up with your family with a clean conscience. If they know you are making the effort, this might even lead to a conversation about how they can get involved and lead more sustainable lives.

Sure, it might be 'more sustainable' to avoid flying. But sustainable living should be less about sacrificing and more about... well, living. So, how can you 'justify' a large carbon output such as flying?

Carbon Offsetting

If you're worried about the carbon effects of flying and can't reduce your daily carbon footprint enough to justify a once-a-year round trip, you should really look into Carbon Offsetting (as initially covered by @HighlyIrregular's answer).

A relevant example which may help put things into perspective is Tom Scott, a media personality and YouTuber (covering tech, science, history, culture, and nature). Tom flies all over the world to film on-location for his various series. Here's what he had to say in regards to carbon offsetting his travel:

The best solution would be to reduce my carbon footprint. Which I intend to try: but it’s difficult when my job requires me to travel to lots of places, and I don’t want to give up my job. Outside of travel, I don’t have much of a carbon footprint to reduce.

So: second best choice. I can offset. Or, more accurately, my company can.

Based on recommendations from folks I trust, I picked a carbon offsetting service, one that:

• supports sustainable projects around the world, so there’s an actual, short-term impact that directly helps people now
• and ensures there is genuinely a long-term reduction in emissions.

That way, even if one of those approaches doesn’t work, hopefully the other one will.

He then goes on to make the point that - if you can afford the cost of flying, you can afford the cost of offsetting. 1.3 tonnes of CO2 should run you about $15-$30, judging by the projects on the site Tom linked, which is not a lot of money when compared to the cost of international flights.

So whether you fly regularly, or only once a year, you should consider factoring in the cost of an offsetting service.

Choose an appropriate airline

There are also some airlines that offer some form of carbon offsetting at the time of booking. Consider booking with these airlines over those that do not offer any carbon offsetting - as this may act as positive feedback, allowing them to continue to offset, and/or put pressure on other companies that do not.

_{Note: I have no affiliation with Tom Scott or his company I just found him a relevant example of someone that can't avoid flying frequently}

I recently returned to NZ after visiting family in the UK for Christmas. This is a round trip of some 40,000km. My solution is to pay for some 60 trees to be planted, here in NZ. This is not a perfect solution, I know, but it salves my conscience somewhat and is definitely better than doing nothing. IMO, Tim’s answer above is accounting sophistry that is used so that nobody takes personal responsibility for their carbon emissions. It’s tantamount to the tragedy of the commons. i.e. it may be true for the individual but it breaks down when everybody applies it.

If you buy that plane ticket, it will go to the "more fossil fuel economy". That's how the CO₂ figure is important.

Then an argument would be to say that you are saving the money for the flight to invest it into environment protecting projects, e.g. like try to save India from mass thirst or into sustainable energy production projects.

Caring for the environment is also caring for your family (however less for the elders, that's how I understand it is difficult for them to understand).

Carbon offsets aren't a perfect solution by any means, but provided the activity being funded by the offset is genuine and sensible for the long term, carbon offsets could certainly be considered to be a good solution.

For example, if you help fund technology that reduces fossil fuel use that wouldn't otherwise have been able to be funded, then you could argue your net fossil fuel use has been reduced. When I did a Google search for carbon offsets, it looked like good quality results were found.

"1.3 tonnes of CO2" is a figure that someone came up with by averaging the fuel per passenger trip and then converting it to emissions. It's not a number you actually need to worry about.

Think about it this way: A flight from A -> B will depart at the scheduled time regardless of how many passengers are on board. Virtually the same amount of fuel will be consumed regardless of the number of passengers on the plane because the mass of the passengers (and their luggage) is only a small fraction of the total mass of a loaded plane. For example, the ~266 passengers (and their luggage) make up only 17% of the mass of most Airbus A 300-600 flights.

So, whether you board the plane, or sit at home, the same amount of CO2 is going to be released into the atmosphere over the next few hours as the plane flies from A -> B. The airline will not cancel the flight just because you decided not to board. 1.3 tonnes of CO2 will not be prevented from going into the atmosphere every time you refuse to fly.

A single additional person adds a trivial 0.064% to the emissions of a given flight. For a 2,000km trip it's roughly on par with a four-cylinder car with one extra person inside. So, unless you object to people ride-sharing in four-cylinder cars over long distances, you actually have no reason to object to being a passenger on a long-distance flight.

Remember: The plane will still fly regardless of whether or not you are on board. 99.936% of the emissions will still be released, regardless of whether or not you are on board. You may as well make use of inevitable/unavoidable emissions and go visit your parents.

PS: I just calculated some more specific numbers for the opportunity cost of a single passenger on an A 300-600 over a 2,000km trip. Assuming the plane is half-full of just about everything (fuel, cargo, passengers, etc.) it works out to be ~26L of aviation fuel. That's it. You getting on the flight will only add 26L to the amount of fuel consumed by the plane over the entire trip. That's about 69kg worth of CO2 emissions — a far cry from the 1.3 tonnes you are worried about. How far can you drive with 26L of petrol in your car?

PPS: A commenter asserted that the above logic could result in the airline scheduling additional flights. That line of reasoning only holds true if you try to get onto a flight that is already at or near capacity. If you take advantage of a flight that is up to 50% booked, no manager in the world would use your booking to justify more flights. Wouldn't happen at 60% either. Around 70% you may be starting to generate a bit of pressure. At 80% you probably would be generating pressure. Above 90% you definitely would. So... don't do it. Avoid flights that are already near capacity and you avoid exerting any pressure whatsoever for airlines to increase the number of flights servicing a particular route. Catch a flight with a low number of passengers and not only is the issue avoided entirely, but you also do the environment a favour at the same time (because opportunistically flying 2,000km results in far fewer emissions than driving the same distance in any sort of petrol-powered car).

PPPS: All of the above refers to long distance international flights. It is almost unheard-of (in modern commercial aviation) for scheduled international flights to be cancelled due to low passenger numbers — the plane, crew, and cargo still need to get from A -> B because they are already booked to go on to C -> D -> E... after that. Short-distance (<1600km) domestic flights are a different matter entirely. Some of the airlines that service such routes can and do consolidate and cancel flights if passenger numbers are low. If you refuse to fly such routes, there is a small probability that, one day, you might actually be responsible for a plane not taking off. Math supports a strategy of reducing emissions by minimising short-distance domestic air travel — it does NOT support a strategy of boycotting long-distance and/or international air travel.