It sounds like you are describing two different problems.

FIRST PROBLEM: Turning off a hot water heater.

First, there are two ways to hook up two water heaters: series and parallel.

SERIES

• In series, the water comes in to the first water heater, then passes through the second hot water heater, and finally goes to your fixtures.
• If you turn off the second water heater, then the water in the second one cools down between uses. This results in very long warmup times.
• If you want to turn one of these off, turn off the first one. (This is actually a legit way to save on energy costs if you don't need the extra hot water volume because in many places the water supply is colder than room temperature and the first tank allows the water to sit and reach room temperature before it's heated.)

PARALLEL

• In parallel, the incoming cold water is split between the two water heaters. Hot water coming out of the two heaters is combined and sent on to the fixtures.
• If you want to turn off one of the water heaters, you also have to close the valve to the inlet or the outlet. Otherwise you will just be mixing the cold water in one water heater with the hot water from the other heater.
• Leaving one heater closed off for a long time is a bad idea because the water can become stagnant and foul.

SECOND PROBLEM: Pilot light going out

Just turning off one heater should not cause the pilot light on the other to go out.

I would surmise that your heaters are connected in series and that is causing surges when one unit is left powered off. This is because of the unbalanced state of heated vs cold water. When water moves from the cold to heated system the heavier cold molecules are compressing the heated molecules that are oscillating in an excited state. This is causing the pilot light to go out as a result. This is known as the Bernoulli principle that is rarely seen in liquids but it was identified as a interaction routinely seen when two disparate bodies of water collide in a circular reservoir. To avoid this you should eliminate the unused tank altogether or regulate the temperatures in both tanks to be within 15% variance of each other. I hope this helps.

My guess is that the chimney for the two water heaters was sized to provide draft for two pilot lights, not one. The air isn't getting hot enough to push (or pull, depending on your point of view) the warm and low oxygen air out the top of the chimney. Instead of warm CO2 floating up and away it sits under the water heater until there is no longer enough oxygen to maintain the flame. With two pilot lights going there's enough heat to the chimney that a sufficient draft can be maintained. With oxygen depleted air drafted up and away free oxygen rich air can be brought in to keep the flame burning. Wind speed and direction at the top of the chimney can affect the level of draft.

That's a theory on why you are seeing what you are seeing. There's a handful of means to address this.

First is to simply keep both lights lit. If they don't go out when both are lit then keep them both lit. The concern, besides the pilot going out, seems to be a desire to keep heating fuel costs down for hot water. How this can be done depends on the water flow path.

A series connection means that there is one tank feeding the next. In this case simply set the temperature lower than that of the second. This is then feeding warmed water to the next so that it will not have to take as long o heat the water to a comfortable temperature. I would take care in this as water that is not hot enough will encourage bacterial growth. If the minimum safe temperature is already close to the desired temperature of your hot water then there may not be much savings. If the water heaters are in parallel then both water heaters would have to be match in temperature anyway.

I'm no professional HVAC installer but I've seen odd things like this when helping my brothers work on their rental properties. The easy solution is to keep both water heaters lit. The harder part is checking the chimney for proper size, height, etc. It may be that the chimney is not quite up to spec but with the heat from two pilot lights it keeps the proper air flow in spite of this.

Check if there's something that might blow enough air under the water heater to cause a problem. I can recall a case of a pilot light going out because of a floor fan, used to dry a floor, would have the moving air diverted into blowing out the pilot by people stepping around the fan. Getting the pilot lighting hole covered (left open to aid in re-lighting) and moving the fan fixed that problem.