Hello.
That's how the parking lights are supposed to work. Well, except for that not flashing thing, but, other than that, that's how they're supposed to work.
The bulb on the parking lights is a single-filament bulb, so if it will come on at all, the problem isn't the bulb itself or the bulb's ground. Your problem is in the wire that runs from the turn signal switch to the bulb, or the turn signal switch itself. There is a connector a little ways downstream from where the turn signal switch wires come out of the steering column, kind of near the pedals, that has eight wires. The one that powers the right front turn signal should be white with a blue stripe, I'm pretty sure. You can take a volt meter and see if that wire is getting any power when you hit the turn signal. Don't unplug the connector because then the whole switch is dead. You can stick the positive probe down into the connector at the correct wire without damaging the wire. If there is no power at that wire when the turn signal is engaged, then you need to replace the turn signal switch, which isn't really that big of a deal. Hope that helps.
1966 Mustang Park Lights. Please tell me how they are supposed to work? 
