Rear parking lights and third brake light wont go off

On most models the brake light circuit runs thru the turn signal switch. The third brake light does not. So it could be a problem with the turn signal switch.
A ground problem would also affect the parking lights.

If you have power thru the brake light switch, the next component to test is the turn signal switch. If the regular brake lights and turn signals are on the same circuit they both go thru the turn signal switch.

have a look for loose or corroded connections at the lights themselves.

There is no adjustment. The park lights are on a different circuit than the brake lights.
Do you have a third brake light that works. Its also on a different circuit but works from the same switch. Do your turn signals work. They use the same filament as the brake lights.
Also, the brake light circuit goes through the multi function (turn signal) switch.

Since the brake lights and turn signal lights use the same filament in the bulbs, the brake light circuit will pass through the turn signal switch. When a turn signal is used it cancels out the brake light for that side.

Let me know if your third brake light and turn signals work in the rear.
Also, if youy have not already, check/change the bulbs. The type of sockets those lights are in also get burned so look at them well.

Is the third brake light working the one up top? If not get a brake light switch check fuses and if the is get a head light switch sound crazy but yes and check circuit boards the bulbs pop in and out of look for tan goop in the sockets

Since the high mounted brake light works you know that the brake light switch is working. Now you need to check the brake light bulbs in the rear tail light assemblies. The brake lights are separate from the tail light (parking or running lights). There are two screws that hold the rear light assemblies in place.

The parking lights and break lights are different. You need to look at the bulbs, (Does it have a third brake light in the rear window that works?) look at fuses, and also you have a brake light switch located on the upper side of the brake pedal arm.

Not even the third eye? then the fuse is blown or all the bulbs are out or the brake light switch on the break pedal is bad wire off or broken( it breaks right off the plug then crosses then blows the fuse as the loose broken wire hits the brake pedel arm. Test the break pedal switch first if the third eye doesesnt work either. no power with a test light the fuse is blown or the wire broke ( the wire goes straight to the switch from the fuse and doesnt go anywhere elese. the other wire if hot should light the lights. Most of these wont come out of park without this working as the wire also turns on the park release but some Fords the park release sticks released and Ford made em that way as they didnt like the brake to get out of park law. If the brake lights dont work maybe you ought not be able to get out of park anyway till its fixed. the switch is hot on both wires with the switch pused down and the lights dont still work go after the third eye first if the bulbs are bad change em before you do the rest of the work. on cars with yellow turn signals the brake lights all work on the same wire but red only use three wires, but you didnt say the turn signales didnt work if they work and they are red you r on the way to the turn signal switch as the brake switch powers the turn switch with power then it splites it fot the two lights and adifferent switch circuit for the front hooks them all together for hazard. yellow turns in the back or even separt bulbs can mean blown bulbs ther too its old enought for all the bulbs to fail and the runs still work.

Brake light relay is bad. It is the third solenoid from the left on top of firewall under hood. If this is bad, the third (cab top) brake will still work fine, but no rear brake lights!

have you replaced the bulbs?