Dont know if this is normal.

When the lights are off, both brake lights work without a problem

When the parking lights are on or the headlights are on, the right side brake light goes off when you brake, but this happens just for the brake light bulb, I guess its the main one, the rest of the lights work fine. I tried replacing the bulb for the brake light and no change..

Would be great if someone could point me in the right direction.

Check the ground connection for the light assembly. Make sure it is clean and secure.

Definately sounds like a ground problem,each socket has it's own ground wire.