What makes you say the parking lights are coming on?
Could it be the front turn signals instead?
How about the license plate light, does that come on?
OK, if your not seeing the license plate light then what you are seeing is the turn signal lights. They share the same bulbs, just a different filiment. If you were really seeing the parking lights, you would also the the license plate light becaiuse they are all connected. they is really no relationship between the brake lights and the parking lights except at the bulbs anyway.
the problem is in either the turn signal switch or the related wiring coming up the column. If you didn't pinch the wiores somewhere, then it has to be a bad turn signal switch. The brake lights and turn signals come together inside that switch and it's entirely possible to end up with a problem like that.
It's always possible that you could have the wrong switch or the plug for it is wired wrong.
OK, if your sure that light is on, then the only place it can happen is in one of the tail lights. either the plug is shorted or the wires are shorted together somewhere.
There is nothing in the turn signal switch that's tied to the parking lights.
You'll have to prop the brake pedal on and start disconnecting the rear lights and remove the bulbs one at a time until those front lights go out.
If you need a wiring diagram for color codes, I'll need the year, make, model and engine size and I'll see if I can get you one.