I'm having a similar problem. I have a 67 Mustang 2+2 and the parking lights go out when the headlights go on (front only). The turn signals and flashers work, but the parking lights are out. We Ohm'd out the headlight switch and when the switch is pulled all the way out to turn the headlights on, the leg for the parking lights drops out. So therefore, I think that this is the way that it's supposed to be, but I don't know for sure. I talked to a Mustang restoration guy and he said that the parking lights should be on when the headlights are on. According to the switch, the lights should be off (which they are).
I'm wondering, do I have a problem? Or is this Mustang guy wrong.
