Why do my upstairs toilets run randomly?
First thing, replace the flapper; it's cheap, easy to install and probably half if not all of your problem. Opening a faucet just triggers the fill-valve sooner than it would normally try to make up for the slow leak.

Next, try tightening the the screw that holds the float at the correct fill level. I like to see at least a 1\4" before it will spill into the over-flow. Yours may have a fill-level line but don't be afraid to short it some. Be aware that messing with the guts might make it worse. Be prepared to replace the fill valve assembly along with the flapper, which will almost certainly solve the problem (for another 10 years).