After a few attempts. I traced power down to behind the bulbs with the lightswitch out. So the rear running lights should work.
Still no rear running light. Replaced bulb with one from my 1959 , same thing.
Ground should not be an issue cause blinkers and stop light work as they should. If I switch the wires , I have no blinkers and no brake lights but the running light works but too bright.
I don't know.
That sounds like a short in your parking light wire. It sounds like it is grounding through that short instead of the housing.
Check the socket, it might only be shorted when the bulb is in and pushing on the contacts.