Its the taillight circuit board or a broken wire. If you look on here i asked same question and i posted up what the reason for my issue ended up being. Moat likely a bad or broken ground, circuit board that is on the back of the taillight or a broken wire. To figure it out get a test light clean a piece of metal connect teat light ground, then unplug taillight and stick trat light probe into 1 slot at a time see what works and what doesnt and write it down. Then go to the other taillight do samething.
Either you have a broken wire, or blown fuse or you have a messed up blinker switch. The way those taillights work theres a white wire under dash on the column that white wire is for brake light. So the way the brake lights work they go through a relay and through blinker switch. So you could have a blown relay i would start there. 1 relay is on the back of the fuse panel, the other is next brake pedal on that panel to the left of the brake pedal. Try replacing those, and if that doesnt work look for broken wires under dash, and check all the fuses, replace blinker switch, and hazard switch. If all that doesnt fix it then you must have broken wires or bad grounds
White wire off the brake switch and also with the cluster of wires front front to back is the brake wire that turns on the brake lights i just figured it out on my 92 chevy k1500 because i have the same issue and i need a inspection sticker
test the white wire on brake switch it should be the wire that makes brake lights go on run a wire down to back and split it to both taillights and your all set Update (11/11/2015): On my 92 k1500 purple and dark green have power when you press on brake the white does not have power when brake is pressed. Idk if this is normal or not