How do I correct the problem of the park lights and brake lights from staying on after ignition is off?

Two reasons for the brake light to be on. 1) Parking brake and 2) problem with pressure in the brake lines.
If the parking brake is off and the ignition is on but the brake light stays on, then there's a problem with the switch attached to the parking brake. It might be stuck or broken.
If the brake light goes on when you apply the brakes, there is an imbalance somewhere in the brake lines between front and back wheels. This light usually goes out when your foot is off the brake, but if it stays on, the switch in the brake lines near the master cylinder might be stuck or broken.

Parking Brake Warning Light - RepairPal

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• As long as there are no problems detected in the parking brake system, the parking brake light will stay on until the parking brake is completely released.

The ABS light illuminates when there is an "out of specification" condition. From your description, it sounds like your parking brake cable may require tightening, or there may be an issue where the rear brakes are not properly releasing when the parking brake is released. With car that is nearly 20 years old, it is not inconceivable that the rear brakes and parking brake assembly may require regular service.

A good first step would be to have error codes read out by a local mechanic or auto parts store. This may give you a better indication of the specific problem. In any case, ANY brake warning should be taken VERY SERIOUSLY and corrected IMMEDIATELY, as you may have a potentially dangerous potential of catastrophic brake failure. PLEASE GET THE ISSUE FIXED RIGHT AWAY!

sounds like u have a bad brake light socket or the parking light filament in one of ur bulbs has blown and is now touching the brake light filament causing it ti cross power

Sounds like a brake light. Check your brake fluid and fill to the correct level if low then see if it goes out.

The Brake light says "BRAKE" is the Brake System Warning Light: "This light monitors various brake functions, including brake fluid level and parking brake application. If the brake light turns on, it may indicate that the parking brake is applied, that the brake fluid level is low, or that there is a problem with the anti-lock brake system (if equipped)."

Another is the Anti-Lock Brake System Light: "This light monitors the Anti-Lock Brake System. The light will turn on when the ignition switch is turned to the ON position and may stay on for as long as four seconds. If the ABS light remains on or turns on while driving, it indicates that the Anti-Lock portion of the brake system is not functioning and that service is required. However, the conventional brake system will continue to operate normally if the BRAKE warning light is not on."

It could also be the Tire Pressure Monitoring light. Check the brake fluid first and see if that fixes it.

Apply the parking brake with the vehicle parked, engine turned off and transmission in Park. Release the parking brake completely, then apply it again, and ensure that it is fully released. Turn the ignition on, check to see if the Parking brake warning light is on. If it is NOT on, apply the parking brake and check to see if the warning light comes on. If the light come ON wihen the Parking brake is applied,.....fully release the parking brake and check if the warning light goes OFF. If it does NOT go off, then an adjustment is needed at the parking brake switch.

Do you have a separate parking brake light and a "regular" brake light? Most vehicles only have one light, it illuminates when the parking brake is set, but it also illuminates when the system detects a failure anywhere in the brake system.

You could have a separate light for the anti-lock brake system (ABS ).

Your description of a light that is on till you apply the brakes is usually a sign of some sort of system brake failure. Start with brake fluid level, then inspect the brakes at each wheel, looking for a leak.

If nothing detected there, then you likely have something more serious. (Master brake cyclinder bad, possible something in the ABS brakes, etc.)

Others may disagree, but I would suggest not driving the vehicle (except to a mechanic) until this is corrected.

bad earth to lamp clusters usually when tow bar electrics fitted - or chaffed wired in lamp feed

not adjusted correctly , sould be able to adjustswitch under hand brake boot

Usually when your signal blinks rapidly it indicates a blown bulb.   Switch on your four-way flashers and check your bulbs at each corner.  On the brake light staying lit, check to see if the left-rear bulb is installed correctly and is the correct bulb for the application.  I've seen people force bulbs in sockets causing the braking element of the bulb to light instead of the parking element.  The braking element is much brighter than the parking element.