I've noticed something similar with my empeg. Ground loop noise when the engine is off, none when the engine is running.

It's still a ground loop, but this time it's due to the change in voltage that happens when the alternator isn't pumping power into the system (I think). Without the engine running, the voltage ends up dropping pretty low. I'll bet that when you turn the engine off and turn the volume up, you also get the occasional blank screen followed by the battery icon, right?