That is very strange that they light up while the key in on assy , but go off when you start the car.... Do they also come on when you just put the key to the run position with the engine off or just assy (accessory ) .
Im trying to think of what is energized while in assy that is not energized during engine operation. Obviously there is a wire some were sending power to them that is bypassing the reverse switch... I cant think of anything that would be energized during assy that wouldn't be during engine run. How ever there are things that energize during engine run that don't during assy, turn signals come to mind because I believe the reverse switch gets power from the same fuse as the turn signals. In assy the trun signals should be dead.
When ever I have an electrical problem the first thing I do is start looking for any thing that someone else has tried to rewire , fix or install... Look for black electrical tape that looks out of place or wire splices that don't belong...