Why would the brake lights stay stuck on in a 1992 toyota when the headlights or just the parking lights are turned on but work properly when no lights are turned on?

park brake light is come on when parking brake is set position. if light stays on when released the switch is located on the parking brake is defective and needs to be replaced, generally not to difficult to accomplish this replacement

low brake fuild or brake fuild leaking
check the brake master cylinder pump brake fuild lever in the engine room .

tail is park, so turn off the parking lights ,.there are 20+ lamps
inside and out for park/tail/dim dash lamps.
do you mean stop lights stuck on, id bet so. (parks out)
so replace or adjust the brake lamp switch.
see alldata.com for full facts on topic.

Park brake lights normally operate by a small plunger type switch the turns off when the level or pedal is placed in the off position. Over time the spring in the lever weakens and placing the lever in the off position no longer depresses the switch. The switch can sometimes be adjusted to make better contact, or the switch bracket can be bent so the lever and switch make proper contact.

Could be a bad brake switch, but not sure since you say brake lights are working. Others reported a blown tail light fuse bit again, since you say your lights work then it may not be the case. I would start by inspecting every fuse. Shouldn't take too long. Then check the brake pedal arm to see the switch. Maybe there is more than one? Or, if the gear selector lock rus off the same 12v as your brake lights, maybe there is a broken wire or the solenoid that locks it is bad.

Check the fuse?

Yes, transient current. Typically happens with damaged taillight bulbs, or poor ground. SO, when you step on the brake, rather than the current going to ground after the stop element, it goes into (and powers up) the running lights, of which dash lights are part of....
Look for ground issues, run external ground if needed, for testing purposes..

In Canada it's mandatory to have DRL(Daytime Running Lights). Your Corolla probably has this feature built in. it is a safety thing however I can understand you wanting them off if it's not the law where you live.
In our 1998 Corolla, I use the parking brake all the time - every time I park - and I've noticed the lights do not come on until I release the parking brake after starting the motor. When I park the car, until I shut off the engine, the lights stay on regardless of the parking brake status however, if I shut off the engine and restart with the parking brake on, the lights do not come on.
If you "adjust' the parking brake switch(i.e.: make e-brake light in your dash it stay on all the time) your lights will not go on until you turn them on via the multifunction lever(left side of wheel). This of course negates the benefits of the e-brake light working correctly (warning you that your e-brake is on or that your brake fluid level is low) which in itself is not so safe!
There is a fuse/relay (I think) in the big fusebox under the hood that has a DRL "spot", you might want to check if you can remove it and still turn the lights on manually when you want them on. Alternately, there is a fuse in the interior fuse box(in Canadian models anyway) labelled DRL. When that fuse is out, it seems the lights only turn ob when I turn them on manually.
Hope this helps and God bless,
Gilbert

It sounds like your brake switch has shorted out.It also controls being able to get your car out of park as well.your foot has to be on the brake pedal to move the shifter and the switch is probably bad.