Why does water come out of the hot supply line when I open the cold?
My sink has a hot faucet on the left side of the sink.  It also has a cold faucet on the right side of the sink.  If you want warm water, you have to mix it in the basin.  

If every faucet in your house is this way, then your problem is impossible.  

However, I suspect you have a valve newer than 1930, with a single spigot. Which means there is a chamber before the spigot where the hot and cold water merge and/or mix.  At the risk of stating the obvious, this means the cold and hot valves are in communication.  

Water can flow from the cold pipe system, through the cold valve, into the chamber, through the hot valve, and into the hot pipe system.  Yes, it's supposed to flow out the spigot "instead", but realistically it will do both.  Pressurized water will seek every path at once, like electric currrent. 

Normally, the higher pressure in the hot-water pipe helps seat the valve.  But here, pressure in the chamber is higher.  No telling what'll happen.

This is more likely to be a problem on more modern valves like a Delta or one of those horrid temperature-control valves found in hotel showers. Or how about a washing machine?  Do you have any users who leave the shower valves at their preferred temperature and shut it off at the showerhead?  (I assume someone on the planet does this.)  Any of these can cause this behavior.   In a perfect world, every faucet valve would include an anti-backflow check valve. They don't.