Its most likely a bad ground. That is a tell tale sign if the indicator light is staying on.
Get a piece of wire and touch it from your vehicle frame to the socket for the bulb and see if the light goes out on the dash.
The tail lights ground through the body not in the wiring like the front lights do.
'87 YJ, 4.2L built and bored, 8" of lift, 8.8 rear, AX15/NP 231J, Tom Woods drive shafts, Flat fenders, 12,000LB winch, Custom bumpers, Rattle can paint job. Never completed...