You didn't say what year model you have.
1. If you have pre 1996 then it sounds like a bad ground connection on the rear lights or perhaps some past bad wiring repairs. The parking lights are possibly reaching ground thru the brake light side of the bulbs. If this is the case then the brake light filament will glow with the parking light side because current is flowing back thru it.
If you happen to have a 4-way trailer plug that a previous owner perhaps "wired" in, that would be a good place to start.
Otherwise start checking for a bad ground. You can remove one bulb at a time and check to see if it's related to one side or the other.
2. Many of the later models (1996 thru about 2000) GM's had rear wiring harness problems from the factory. My company had GM service vans during that time period and almost all of them had problems to some degree. They would exhibit the same problems you relate.