remove the license plate and reach up through that opening. Corvettes weren't designed to be easy to work on.
Sounds like it may have been replaced at one point with an inferior piece.
Change brake light bulbs Just remove the rear license plate. You can change the backup lights and brake light easily.
I found under the drivers dash on the brake pedal the brake light switch to be faulty. It can operate the tail lights but not the center brake light replace it. This was the case on my 90 vette.
Change the bulb and check to see if it's not grounding out.
back side of the brake pedal
just above the brake peddle
underneath the brake pedal
== == how do you reset the oil light on a1992 corvette
It is on a bracket, just above the brake pedal.
remove tail light and reach through
It is mounted on bracket just above the brake pedal.
It is under the dash, just above the brake pedal.