The 1968 model year was the first (on cars) where the parking lights stay ON when the headlights are turned on. The 1967 and earlier cars do NOT stay on with the headlights. The 1968 deal was a then-new Federal requirement, and applies to all 1968 and newer cars sold in the US. This affected trucks, starting in the 1970 model year.
This is where the terms "parking lights" and "driving lights" came from. It was considered impolite to have your headlights on while parking on a street or in a parking lot, so as not to shine lights in the other drivers' eyes. You use your "driving lights" on the street.
As far as the issue on your car, it sounds like a bad ground or bulb at the tag light.