What is it called when you raise a problem and someone else makes it seem like you're entitled by bringing up something far worse?

This is the fallacy of relative privation, which basically argues that you don't really have a problem because there are other people suffering from a much more severe version of the problem.

Here's a quote:

The fallacy of relative privation, or appeal to worse problems, is an informal fallacy which attempts to suggest that the opponent's argument should be ignored because there are more important problems in the world, despite the fact that these issues are often completely unrelated to the subject under discussion.

"You raised a valid point, but now someone has completely dismissed it by bringing up something unrelated."

This looks like an example of a

false analogy (also questionable)

It's a Faulty Comparison:

If you try to make a point about something by comparison, and if you do so by comparing it with the wrong thing, then your reasoning uses the Fallacy of Faulty Comparison or the Fallacy of Questionable Analogy.

One-upmanship, perhaps - The art or practice of gaining the advantage, the condition of being ‘one up’ (OED); the art or practice of successively outdoing a competitor by discomfiting them (Wiki)

Ridiculously exemplified by Monty Python - https://www.youtube.com/watch?v=ue7wM0QC5LE

Whataboutism

To answer the question directly with a single word, I would say this is an example of 'whataboutism':

The practice of answering a criticism or difficult question by attacking someone with a similar criticism or question directed at them, typically starting with the words "What about?"

(From Cambridge)

To rephrase your original examples:

"These working conditions are subpar. We don't get any breaks." and then someone else says "You're entitled. What about people who work in sweatshops!"

"I'm hungry." to which a parent might reply "You're not hungry. What about the starving children in Africa?"

An ad hominem argument

Free dictionary:

1. Attacking a person's character or motivations rather than a position or argument: The candidates agreed to focus on the issues rather than making ad hominem attacks against each other.
2. Appealing to the emotions rather than to logic or reason.

Usage Example An ad hominem argument is a personal attack against the source of an argument, rather than against the argument itself. Essentially, this means that ad hominem arguments are used to attack opposing views indirectly, by attacking the individuals or groups that support these views.

Your conversation partner might say that you have a "First world problem".

"Weird Al" Yankovic wrote a nice song about this phrase. However other people are more critical about the use of this phrase and find it often insulting and condescending.

I've heard it called cause hijacking. Somebody has a problem (their cause) and then somebody else dismisses their problem by bringing up a a bigger problem.

If all the support and energy behind a cause were an airplane, it is as if somebody hijacks that plane to go to another destination.

A relatively new expression used to describe this is gatekeeping misery.

The metaphor is that there's some kind of special club for the truly miserable, and the gatekeeper doesn't think you're good enough to get in.

Several other fallacies may fit, depending on how the reasoning goes:

Red herring – introducing a second argument in response to the first argument that is irrelevant and draws attention away from the original topic. In this case, the original argument is made about poor working conditions in this particular company, and the distracting argument brings up poor working conditions elsewhere.

Appeal to emotions – bringing up emotionally charged facts (starving children, sweatshops) which serve to distract from rational consideration of the original argument.

False dilemma: telling you can either be concerned about your own conditions, or conditions of people who "have it bad", ignoring the fact that you can perfectly care about both at the same time.

Fixed pie fallacy (a.k.a zero-sum fallacy), where the reasoning is based on the assumption that there is a fixed amount of wealth in the world, so by asking more wealth for yourself (more food, better working conditions) you necessarily take those things away from someone, in this case, from people in poorer countries.

Brené Brown calls it Comparative Suffering

Unfortunately, one of the things that’s immediately triggered when we go into fear and scarcity is comparison. Comparison and who’s got more, who’s got it better? What are they doing? What’s crazy about comparison when it’s triggered by fear and scarcity, is that even our pain and our hurt are not immune to being assessed and ranked, So, without thinking, we start to rank our suffering and use it to deny or give ourselves permission to feel. “I can’t be disappointed about my college graduation right now. Who am I to be sad that I’m not going to be able to have this great ceremony, because there are people sick and dying?” Or, “I can’t be angry and afraid about being sick right now, because there are people sicker than me. I can’t be scared for my children because there are homeless kids who have nowhere to sleep tonight. Why should I be tired and angry, I have a job right now and so many people don’t.”

The entire myth of comparative suffering comes from the belief that empathy is finite. That empathy is like pizza. It has eight slices. So, when you practice empathy with someone or even yourself, there’s less to go around. So, if I’m kind and gentle and loving toward myself around these feelings, if I give myself permission to feel them and give myself some resources and energy of care around them, I will have less to give for the people who really need them. “Like what about the healthcare workers on the front line right now or the grocery shop folks or the hourly… The people who are delivering packages?”

Since the 1980s, I have called this one-downmanship. The earliest reference I have to this is a 1983 Garfield comic https://www.gocomics.com/garfield/1983/10/17 where it is presumably intended as a neologistical witticism. It has, however been used more broadly as in https://www.lexico.com/definition/one-downmanship which includes the definition - The art or practice of being or appearing to be at a disadvantage. Compare "one-down", "one-upmanship". And has been used in psychology https://www.psychologytoday.com/au/blog/sacramento-street-psychiatry/201603/one-downmanship as well. It is, of course, and obvious play on "one upmanship" and may have been invented earlier than the 1980s by multiple people.

The particular usage the OP is looking for is one-downmanship by proxy, in that the opponent refers to another person, rather than themselves, as being worse off. For me personally, this seems within the scope of the intention.

Perhaps it's a Straw Man:

The straw man fallacy occurs in the following pattern of argument:

Person 1 asserts proposition X.

Person 2 argues against a superficially similar proposition Y, falsely, as if an argument against Y were an argument against X.

i.e.

• Exaggerating (sometimes grossly exaggerating) an opponent's argument, then attacking this exaggerated version.
• Presenting someone who defends a position poorly as the defender, then denying that person's arguments—thus giving the appearance that every upholder of that position (and thus the position itself) has been defeated.

The straw man isn't specifically about picking a similar argument, it's about misquoting the original argument as being something else, and attacking that instead as a straw man: https://www.artstation.com/artwork/lOWAe

I'd say it's a change of subject by way of gross exaggeration to imply wanton self pity as the original cause of the discussion.

They are trivializing the problem.

Lexico:

trivialize (British trivialise)

VERB [WITH OBJECT] Make (something) seem less important, significant, or complex than it really is.

‘the problem was either trivialized or ignored by teachers’

‘This trivializes the death of thousands of innocent victims.’

The other answers so far all have negative connotations. A way to describe this approvingly (Sometimes people really do have a sense of entitlement!) is to put things in perspective or give them a sense of perspective.

Definition 2b of perspective at Merriam-Webster:

the capacity to view things in their true relations or relative importance

A much harsher way to put it is that self-centered people should get over themselves.

Non-sequitur. It does not logically follow that your state is related to another person's.

In this case it could also be classified as "concern trolling." They don't really care about hunger elsewhere, they just want to score points on some score-board only visible to them.

First world problem!

This phrase, often used as an exclamation or interjection, is a brief substitute for the second person's story. It encapsulates the fact that there are many goods and services which are deemed commodities by those living in economically well-developed countries (so-called "first world" countries), but which are in fact not widely accessible among the population of Earth as a whole.

This is a newish slang phrase I've heard used primarily by folks under 40 years old, but it seems widely understood. It's primarily used to in reference to problems that are only possible because of newer technologies.

There are many usages of this phrase out there on the web, but not much in the way of description of it; so the above description is based on my own perspective.

Usage

Person A: "My router went out last night, and I missed the new WandaVision episode! I cannot wait for it to get fixed."

Person B: "First world problem! At least you have running water and electricity [for example]."