[TJ]

Seth, before buying any parts check to see if everything is properly grounded. I had a similar thing happen years ago with taillights on a '46. I put on turn signals and when the left light would flash you could also see the right one flashing faintly. Chased the problem for 2-3 days. Finally decided to run an extra ground wire from the taillight housing to the frame on each light. Problem solved.
Go back over you wiring again. Since some of the wiring is faded there could also be a wire in the wrong spot. The dimmer switch should have nothing to do with the parking lights. They are separate circuits. A hot wire comes from the headlight switch to the dimmer and the other two posts on the dimmer go directly to the high and low beam of the headlights.
If you low beams are working I suspect there is a wire crossed at the bundle near the front fender or something crossed at the headlight switch.