What kind of truck is it and what components are in it right now? Factory head unit?
Might be a ground issue but unlikely. Lots of variables here, is the amp tied in through a line out converter? Also could be an issue with the amp too. Is there a direct remote wire ran to it, or is it set up to power up when it senses audio? Take a pic of the amp's control panel (usually on one of the sides with the knobs and switches).
My money is on a factory sound system with an aftermarket amp/sub combo wired in with a line out converter because if it does it when you listen to podcasts which usually have no bass, most line outs are set up to power the amp up once it senses any type of good quick bass regardless if the stereo is on or not. It's a feature that deals with most factory head units not having a remote wire output.
ETA: also post a pic of the wiring to the amp.