Could be the multi position switch in the steering column . That's what I think it's called. It's the turn signal switch. My old jeep tj lights up the left blinker when braking sometimes . I just have to wiggle the turn signal arm and it stops but your switch may be more worn or corroded.
It's a 1957 Chevy 3100..with a built 350.. it has a universal turn signal on the steering column..I did not do the wiring on the truck. have not looked at this yet but,thinking that it maybe link to the rear turn signal, as the fronts are single filament bulb and the rear are a double filament.i'm thinking it maybe tied into the hi side of the filament..