I have a 2005 2500, automatic. I changed a tail light bulb due to it burning out, after I replace that a day later, when I turn on the running lights the reverse lights automatically come on. Doesn't matter what gear the truck is in, or if the running lights or headlights are on, the reverse lights are on. With the lights off, put it in reverse and still no revers lights. I changed the piece the bulbs plug into on both sides thinking one had a short in it. Nothing changed. Anyone else had any issues like this before?

 

The only thing that's common to all those lamps is the ground. I bet you have a ground issue in one of the sockets causing a backfeed, turning the backup lights on when the park lights are on. That would also explain why the backup lights don't work when they're supposed to.