What causes the park brake engage light to come on

This may be caused by a faulty park brake switch ,it is usually located in the brake handle where it mounts onto the chassis. This can vary depending on the make and age of the vehicle. If you can locate the switch ,try spraying some contact cleaner or WD40, into the switch.

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!

Try the parking brake release. On a car that old you MAY have a sticky parking brake cable that's partly engaged. If this is the case, it would freeze up in place and cause the sending unit for your brake light to turn on/off at random. -they use the same light for brakes and parking brakes- If your brake petal has become less firm or "spongy feeling" recently... then it's an issue with the brakes themselves.Also check your brake fluid level.

Reach down and release the parking brake at the same time pull up on the parking brake pedal. Sometimes the pedal gets bumped just enough to engage the switch indicating the brake is engaged.

Get down and look at parking brake assembly, If you see an electrical connector, unplug it, see if that helps? If it doesn't help or causes other issues just plug it back in.
Of course that isn't the proper fix. If something wrong with parking brake assembly, get it checked out.

It's possibly a micro switch does the car give a warning beep if you try to move the car with the parking brake on?is the car a manual or auto?

cold weather cause metal to shrink more likely brake drums rusted causing brake drum lockup up.rust acts like sand paper making brake drum surface rough have brake drums turned or use 120 grit sand paper just sand out rust sand direction drum rotate.

Check your brake fluid. The same light is used for the brake fluid level as well as the parking brake.

This light can be caused by your parking being on. If not, it's most likely caused by low brake fluid. Check the fluid level under the hood and fill it up if needed. If that doesn't get rid of the light, you should have it towed to a dealer and don't drive as your brakes may fail at any moment without notice

Check to make sure your parking brake is not partially engaged. If the parking brake is engaged even to one click off from the released position the DRL module will be disengaged. Also there may be a problem with the parking brake position sensor which is responsible for determining if the parking brake is partially engaged.