I have a 1967 Holiday Cutlass Supreme. When I turn the right turn signal on the left brake light goes off. Brake lights work fine with no turn signal on. Right brake light works fine when left turn signal on. Any insight on where to start to correct this issue?

 

Check your grounds for your rear lights.

 

That's the first thing to check, specifically at the left socket, but I suspect it is not the issue here, as the brake and turn signals use the same filaments. Do verify that the symptoms are exactly the same when the tail lights are both on and off.

If they are, then the only way the RIGHT turn signal could affect the LEFT brake light is a problem in the turn signal switch. The turn signal switch includes a set of contacts that switch between the brake pedal switch and the turn signal switch when you use the directionals. If this is bad, it would cause the problem.

 

I would do all mentioned above and also check the wiring over from the ribbon plug at the back set to the right taillight. Many over the years put hitches on these cars and tapped into the wiring with all likes of methods. and might have spliced it incorrectly somewhere in that harness. Might find the wires crossed.

Just a thought
Larry