[71GMC_3/4T]

You have a bad ground in one or both of your taillight housings.
Pull the lens caps off, then remove the housings. Clean any rust off the mating surfaces. When you put them back together, see if you can get an external tooth star washer at the mating surfaces.
If this doesn't work, maybe you have a whole bedside that is making poor contact. In that case, I would run a wire from the taillight housing chassis down to the frame.
Also, clean your bulb sockets with a wire brush. You can get one that's almost perfect for the job at the hardware store. Buy the one for cleaning copper fittings for sweat soldering. Make sure the lights are off when you clean them and blow out any debris with compressed air.