Both back brake lights burned out at the same time

For the lights on the front and dash lights coming on when you push the brakes,check the brake light bulbs and wiring at the back.A defective bulb may be energizing the parking light circuit by transferring voltage from the brake light circuit.If you have trailer wiring this could be the source also.Do you have a sub in the car?They can shorten the life of alternators as they can use lots of power.Have you checked the voltage the alternator is producing and amps the vehicle is asking for when running?

some bulbs have dual filaments in them. One for tail lights and the other for brake/blinker lights.
Many times one element will burn out but the other will still work.

Try replacing the bulbs.....assuming you have already checked the fuse for the brake lights. (many cars have the stop and back up lights on the same fuse)

The caliper is not returning or releasing all the way, reason for burning. Clean and lubricate caliper. Light staying on, air in the brake system. It needs to be thourouly bled

it can be a few problems but i would start by checking the fuses. after that look under the dash where the brake pedal is. you will see a switch there that toggles on when you apply the brakes. i have to mention one other thing because i've had experience with it. check to make sure the bulbs are good. unless your car is equipped with a warning system or you check the brake lights often they may all be burned out over a period of time without you knowing it. a small trick to that is whenever you are backing up or near a something that you can see the back of your vehicle... apply the brakes... if you don't see lights on one or both sides it's time to start changing bulbs.
the tail lights working is independent of the brake lights as generally if it's the same bulb it will have two filiments.

Any competent repair shop with a scanner should be able to pull codes from the ABS computer. It is likely a speed sensor or wire.

This is just a guess , so please don't give me a bad rating. to me it sounds like the switch on the brake pedal is broken ( stuck on ) so brake lights are on , but have burned up the rear bulbs. Highmount light might be LED so runs cooler and does not burn out. Try new bulbs in the tail lights.

Make sure the brake switch is working. Leave it hooked to the wire harness and work it by hand while someone checks the lights. If the switch works it just needs to be repositioned on the pedal and adjusted. If it doesn't work, check the fuse. If the fuse is burnt or burns out again you have a burnt wire. Start tracing them out one at a time looking for a burn mark. I'd start with the power wire first. It's usually red or orange,it's the most likely problem. Cut out the burnt piece and splice it back together. Tape it up well and look for anything it could rub on to prevent this from re-occuring. This should get you back on the road.

Check the bulbs, they may have burned out one at a time undetected.

On the Z cars, that is an indication of a brake lamp that is burned out. Have someone step on the brakes while you look at the back of the car to identify which bulb is burned out.