When our one toilet is flushed, the other toilet will run for a short time? II have drained the tank and cleaned around the flap.
Sounds like a problem unrelated to the flapper. The "running" you hear is likely the fill valve. Perhaps the fluctuation of your water supply pressure when one toilet's fill valve opens is causing the other toilet fill valve to momentarily leak-by or open. I have seen this happen before and a rebuild of the fill valve washers (or fill valve replacement) should fix it.
Try shutting off the supply to the suspect fill valve and open it up. Inspect the valve washers for debris, lime, scale, mineral deposits. Flush the valve by cracking open the water supply while it is disassembled. If the valve washers are worn or soft (if you get black stuff on your fingers when touching them) they need replacing.
I agree with jimmy that it is the fill valve and not the flapper.
The problem is one of water pressure.
The fill valve has a float, and this applies pressure to close the water flow into the tank.
The idea is that the valve fills up the tank to a certain level of water, and will stop.
This is almost, but not quite, how it works.
The tank fills until the float has enough strength to close the valve.
The water stops when the cold water pressure is the same as the strength of the float to stop the water.
When you flush the first toilet, it makes the cold water pressure fall. This is the same thing that happens when you flush a toilet and the shower gets cold.
Like Jimmy says, because of debris and bad washers, the valve can open a little, just enough that the water runs for a few seconds.
Unless you ignore it, then there is so much debris, or the washers as so worn, it will run all the time. Because the float is never gets as strong as the water pressure.
If the above does not cure the problem, it may be that the vent for the second toilet is blocked or non existant. The water draining from the first toilet will cause a vacumn in the drain line of the second toilet causing that's toilet water seal to be sucked down the drain resulting in the start of the re-fill cycle. It generally will not be a flapper valve problem. Does this happen to both toilets? In other words, does this happen when either of the toilets are flushed, does the other toilet "flush?
Is it old? A modern diaphragm style valve might fix that, as it uses the actual water pressure to seal the valve; not just the wonky operation of a float on a stick. It does have a float, but it only works a very tiny valve that works the big one. Kind of like a touch lamp vs a pull chain light. The reason could still be a very slow leak at the flapper, the flapper seat if that assembly is removable, or around the fill valve that evaporates or runs down the tube through the hole in the floor. Mark the full water level with a crayon, check back later