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Are all of the parking lights staying on or just the ones in the rear of the vehicle?
Is it possible it is the brake lights that are staying on?
Ok, that makes MUCH more sense. I was looking over the wiring diagram and would have had a very difficult time seeing how this could happen.
Try removing fuse #23 in the power distribution center. This is the fuse for the brake lights. If they go out then check the brake light switch. It may have come loose from the brake pedal assembly or it could be stuck closed. If the fuse turned off the lights then plug it back in and disconnect the brake light switch and see if they turn off.