My battery light is on only when key is off. It never comes on when motor is running, or when the key is in the on position.

Some posts say bad alternator, bad fuse, or bad ignition switch. Have not tried any fixes yet.

It is just strange that the light is on only while sitting and never while running.

The idiot light is discharging my battery. Why?

 

battery

 

Probably due to bad/failed diodes in the alternator. Diodes are used to allow current to pass in one direction. If it failed, the current can go the opposite direction, thus your battery drains when the car is off.

 

Replacing the alternator fixed the problem.