1993 toyota corolla park/running lights come on after parked.

Check all the fuses and relays that deal with lighting. It may be that the switch is faulty.

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Fuse box diagram Toyota Corolla ... - about fuses and relay

https://fuseandrelay.com > toyota > corolla-e120-e130
Designation of fuses and relays Toyota Corolla E120 E130 (Toyota Fielder) with photo examples and box diagrams. Cigarette lighter fuse.
Mar 30, 2021 · Uploaded by Fuse and relay

park lamps are dimmed to dash illimunation lamps
so park is bad. Precursor.
the park has a fuse and relay on many cars.
check fuses, first. as always forerver and ever, electical

I found this useful diagram at troubleshoot vehicle.com. It answers your question. Fuse box is located below steering column.
Driver Side Fuse Panel
(1993-1996 Toyota Corolla) December 21, 2013 Updated: December 21, 2013 Written by: Abraham Torres-Arredondo In the table below you'll find the fuse location and description of the fuses of the driver side under-dash fuse box on 1.6L and 1.8L 1993-1996 Toyota Corolla.
The fuse box is located on the driver side of the dash, below the steering column.
1993-1996 Toyota Corolla Driver Side Fuse Box Fuse Fuse Name Amps 1 Cigarette Lighter, Digital Clock, A/C System, A/T Shift Lock, Alarm, SRS Airbags 20 2 Tail, Parking & License Lights, Instrument Panel and Daytime Running Lights 15 3 Fuel Injection, Cooling Fans, Battery Discharge Light 10 4 Cruise Control System, Daytime Running Lights 10 5 Fuel Injection System 7.5 6 Stop Lights, Cruise Control, Shift Lock System, Anti-Lock Brake (ABS) System 15 7 Turn Signals, Emergency Flashers 10 8 Starting System, Anti-Lock Brake (ABS) System, Shift Lock System 15 9 Gauges, Reminder Lights, Back-up Lights, A/C System, Daytime Running Lights 10 10 Front and Rear Wiper/Washer System 20

you blew a fuse. check fuses in both fuse boxes..
ahh yes, the IP cluster lamps are dad, but not the idiot warning lights
only the parking lights in the cluster are dead. right.?
and the park-tail, running lights and dash dim lights are all the same.
my guess three is a short on the TAIL fuse, the TAIL fuse blew
and took out: park-tail, running lights and dash dim lights are all the same.
it blew. most are marked tail even thought it runs side lamps and front parking lamps... TAIL is wrong, they are parking. lamps it truth.

may be you should make a diagnostic and you can do it in a garage or you can do it by yourself if you want to do it by yourself you need some tools and you can find some good ones here: diyobd2.fr

If you replaced the solenoid that locks the shifter in park, the problem could be the brake light switch on the brake pedal. The pedal sends power to the solenoid to unlock the shifter.

May not be any fuse and if it was other things would not work either. It is most likely the sitch that controls the running and dash board lights.

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

faulty relay or wire short to ground

take out the 2 phillips screws that hold in the cluster.there are some bulbs in the back of it,look and see that they are all ok.
the tail lights operate on a separate circuit,but the headlight switch could be bad, as it operates both of those sets of lights in the park and lights on settings.check all the fuses and relays in the fuse box under the hood and lower left kick panel