What year???
On PRE-1968 cars, the parking lights up front ARE suppose to go out when you pull the headlight switch all the way out (turn on the headlamps).
First things, first, have you done ANY work up front, changed the front clip for restoration, or pulled out the radiator? Lights need good grounds, any upset grounds and you'll have problems.
Second thing, start stomping on the dimmer switch a couple times. Those things DO go bad, and that controls ALL the power to your headlights.
Next, take some voltage measurements with a voltmeter up front (at the headlight sockets). Green wires should have 12 volts when high beams are activated, tan wires should have 12 when low beams activated. Take measurements with the red test probe on the wires, and the black probe on a GOOD CLEAN GROUND.
If no voltage, I'd try to investigate the dimmer switch on the floor before you go to the work of checking out the light switch.