Going out on a limb here.... when a switched wire like a tail light wire is not powered up, it usually acts as a ground.... when you turn on the headlights and the tail lights are on, the ground for the other lights is probably lost. most likely a wiring issue which resulted in the ground being leached from another circuit. very common.
Do you have access to the wiring diagrams for the stop/turn lights as well as the running/parking lights?
I don't remember if the 96 has the lamp out monitor system, if it does not then you can just make a new ground for all of your lights and then verify that all the power wires are giving a good clean signal.
I have seen lots of people over the years wire things up to a wire that showed a good strong ground, only to find out later it was a power wire but was being fed a ground through the load device it was meant to power. (headlight are really common place for wiring mistakes because when they are on you see 12 volts on the supply wire, but when they are off, they make a solid ground.
years ago we used this to install starter kill switched in cigarette lighters and power mirror buttons.... normally when a car has the key on those circuits are powered, but on some cars, when cranking they fall to a ground. use a relay that has a constant 12v power and is switched by a ground that is only a ground when cranking and the relay will close letting you start the car, once the key goes back to the run position, the relay gets 12v on both sides and opens back up again.
I would start by taking the ground wire off one of the stop/turn lamps and just putting it on a good chassis ground and testing it out. if it starts working then you know the ground was tied to the switched tail lamp power and it was getting a ground through a bulb.
