On the 82 Pontiac, a year a ago the right blinker light remains on in the dash. It's not brighter on the outside. If I put the right signal on, then it will flash on and off.

The parking light in the front right does not work. If I switch bulbs with the left side, it will work for that drive. After that it's back to not working.

Where do I start to fix the problem?

 

Have you checked your right blinker socket for corrosion or wear. If it checks ok check for power to one of the pins with parking light on (might be a bad connection in socket)

 

Thats good. Corrosion is what F'ed mine, looked fine. I replaced the bulbs and cleaned the connections and worked fine. Also remove the housing inside vehicle and clean off corrosion, grease, dust, etc.