I would agree with a bad ground. The tailight bulbs have two filaments. With a bad ground and the parking lights off the positive feed from the brake light switch will use the other lights in the system as a ground and light up the parking light filaments in the tail lights. When you turn the parking lights on it back feeds through the larger filaments to ground. Step on the brake and both sides become positive and out go the lights.
69 RS 355 Tremec 5-Speed 4.11's