What could cause the third (upper) brake light in the rear to come on whenever I turn on the parking lights in my 95 Corolla?
I don't know anything about your car specifically. Generally speaking, if your car uses dual filament bulbs, ie one bulb serves as brake and tail light in the rear, I'd start looking closely at those. Often times the end of one filament will burn or break off in one of these bulbs, allowing it to contact the other filament. That creates an alternate path for electricity to flow, perhaps back feeding the lights you're talking about. Even a bad connection at the socket bulb plugs into can sometimes cause strange things to happen. Other things to check would be the grounds related to your lighting, especially those that might be in the trunk area. Also check the fuses. Sometimes an open fuse causes grounding or back feeding through seemingly unrelated circuits.
That's all I've got. Good luck.
The high mount lamp shares a ground with the lamps on the right rear lens. The most likely cause of it coming on with the parking lamps is a back feed is occuring because of the ground in the center, rear of the trunk failing/corroding.
Pull the high mount bulb while the parking lamps are on and you should see the affected circuits react to the loss of the ground that they are finding through that bulb.
Thanks for your responses Gary & John. I will try to do what you have said. I should have also mentioned that the parking lights (and of course the third(upper) brake lights) come on whenever I press on the brakes, and I hope that this confirms to you that you were right on in your guesses.