You don't say if you are talking about front parking lights or rear parking lights. If front and it is a 70 or 71 it would pick up the ground from mounting connection in the bumper. I would assume the 72 would get the ground from the grill mounting to the car.
Do your turn signal work properly? Since the parking lights and the turn signals both use the same socket and bulb, if the turn signal works the parking lights should work too. If it does I don't think it would be a bad ground. Use an ohm meter and check for you have a good connection from the bulb socket to the frame/chassis or body.
rob