The next thing to check is the switch that activates the brake lights. It is probably on the master cylinder.
Probably Flasher
emergency blinker
Probably the functions, running lights, reverse, brake, blinker, and a ground.
It has been my experience that when the blinker comes on but the lights don't blink there is one or more bulbs burned out. If that is not the case check the flasher.
A simple first step would be to check the fuses to these lights, then go from there.
Fuse for parking lights probably blown.
The tail lights are on a separate wiring than the blinker/brake lights. Look for a blown fuse or a broken/loose wire.
cheack the wireing
your problem could be in the switch its self. unfortunetly they are expensive to replace, i had to actually wire a switch direct to the blinker system.
Bad earth connection at light cluster?
try replacing the flasher. there should be two, one for flashers and one for turn signals
Problem most likely is in the Turn Signal switch. I had same problem , replaced Turn Signal switch assembly and that fixed it.