1967 Dart GT- Brake lights and turn signals won't work when my marker lights are on but work fine when it's headlights on.

The tail/ park/ head circuit should have nothing to do with signals, except

Grounding of the lamps

Possible wrong contact in the sockets, and may be due to something like poorly indexed (manufactured) or just poor quality LEDS

A wiring fault of some sort AKA crossing the tail circuit to the signal/ stop lamp circuit

A defective lamp/ led. ALWAYS remember. "New" does not mean good quality, capable of working, and of course not even the correct part.