Headlight switch continuity?

The parking lights on my 1990 F Super Duty Ford won't come on with the headlights, I've checked the continuity between the parking lights and ground while having the parking lights on and there's continuity, but it's to my understanding when you turn the parking lights off, there should be an 'open line' representing no continuity. But continuity continues to exist in the circuit even when I've turned off the switch. Would this indicate a break in the wire, making continuity exist? If not, are there any other ideas that I could try?

Added (1). The parking lights don't come on at all, I'm talking about when I turn them on and off.

Your understanding that the circuit should go dead is correct. I think you are on the wrong terminal. There will be a hot wire going to the switch at all times, from the switch it is then directed out to relays that will activate the head lights and the parking lights depending what position you choose. I think you are on the hot terminal doing your test, check the other leads from the switch you should see then go off when the switch is in the off position. Look in the owners manual for the locations of the relay for the parking lights and swap that relay for another like relay as a test.

Yeah, just check for 12 volts at the parking lamp and trace back to the switch and battery.

The parking lights come on by themselves, but go off when you switch the headlamps on? That's probably a fault in the switch itself.

You don't perform a continuity test in a powered circuit. I don't know exactly where you were measuring from, but there will always be continuity between the "hot" lamp terminal and ground (unless the bulb is burned out).