Tail light gremlin
if the brake is applied, on the schematic the electric current should follow these paths.
Parking lights on with brake applied.
The details of how the haz switch wires tie in wit hthe brake circuit vary somewhat from year to year.
Lets see what '68 Barracuda shows in the FSM. Or in this case the pdf version done by one of the guys here.
With a little cut and paste to connect the circuits
So we see that the power to the stop lamp switch has nothing to do with the emergency flasher other than the junction.
This is why brakes on will overide the flasher.
Edited the two schematics to remove the e-flasher since it doesn't apply, at least with 1967 and 68