What would cause one brake light to not work when the lights are on?

 

Brake light or tail light? The brake lights only come on with the brakes applied or the emergency flashers going. The tail lights come on with the head or park lights. If one isn't woking then, swap it with the other side to make sure it's not burned out. Beyond that, it could be a bent prong on the plug to the assembly, a bad wire or a bad ground. Usually, a bad ground will cause it to be dim, unless you have LED bulbs and thise are either on or off. They won't work at all with a weak ground

 

What year? Looks like an '01+ in the pic.
They all share the ground. But, each brake light get their power from the combination flasher. It controls the turn signals/brake lights.

 

Clearly stated, brake light.

 

04 3.7 5spd manual

 

Brake lights function properly until headlights switch is engaged. With parking lights and/or headlights switched on the left brake light/tail light shuts off.
No, it's not the bulb and yes, I've switched out the socket.

 

The headlamp switch and combination flasher share a ground. It's a "sense" circuit, low current to trigger the CTM outputs, on both.

Should be on the side of the instrument panel. Follow the harness down from the CTM, it'll branch off.

 

CTM? Central timer module?

 

Yes, Central Timer Module.