You can try this quick check for a bad ground at one of your tail light assemblies. Turn the park lights on which you say also turns on the brake lights. Now simply undo either brake light socket. That should turn out the other brake light and the badly grounded park light (properly grounded park lights will remain lit). This will tell you which side has the bad ground, typically just corroded. Using Oxguard on ground connections (dielectric grease on other wiring connections) can help keep this from happening again.

Even though we normally think of the brake light system as independently divided left/right, because there are no diodes in the bulb integrity sensor (it's just a wiring splitter with current sensors) the left brake light is still electrically connected to the right brake light. So in your case, one park light ground wire is no longer making a direct connection to chassis ground, but the ground wire is tied to the nearby brake light ground wire (also lacking a direct ground). So current flows backwards through that brake light (now lit), up through the bulb integrity sensor under the dash and back down through the other brake light circuit through its bulb (also now lit) and then to chassis ground. I think this accounts for all your symptoms, there are other perverse possibilities, but this seems the most logical. You should notice the brake lights getting a little brighter when the brake is applied. You should also notice that with the lights switched off, at least one rear park light will be lit when the brake lights are on.

Anyone who's spent any time trying to get trailer lights to work properly realizes how bizarre things can get with just a simple bad ground.