How are the exterior lights set up?

Weird question, but my 67 coupe puts the parking lights on when the switch is in park, but shuts them off with the headlights on. While the front parking lights are on, the turn signals blink the second filment (1157na bulbs), but with the headlights on the turn signal blink the same filiment. I have seen pics with mustangs headlights and parking lights on at the same time, and I want that.
 
Pre 70' Mustangs had the front park lights wired to turn off when the headlights come on. To wire them to stay on with the rear park lights you need to take the head light switch connector and cut the wire that feeds the front park lights and splice it to the wire that feeds the rear park lights.
 
If my schematic is correct, the Black wire in the headlight switch connector (terminal "R" on the switch)feeds the rear park circuit and the Black/yellow (terminal "P" on the switch) feeds the front park circuit. Cut the Black/yellow wire and splice it to the Black wire, and the fronts will stay on with the headlights.
 
Weird question, but my 67 coupe puts the parking lights on when the switch is in park, but shuts them off with the headlights on.

This is normal. In fact, it was required by law at the time.

While the front parking lights are on, the turn signals blink the second filment (1157na bulbs), but with the headlights on the turn signal blink the same filiment.

This is not normal. The park lamps use the small filament, the turn/flasher uses the large filament. If they are cross-connecting, you have a ground problem.

I have seen pics with mustangs headlights and parking lights on at the same time, and I want that.
You can do that by connecting, under the dash, the taillights wire to the park lights wire with a Scotchlock. If your park lamp ground is screwed up, though, it won't solve your ground problem.