Parking lights stay on

Do they get brighter when you turn them on with the switch or do they stay the same brightness? Do you have any tow options? Do you have roof markers? Are they on? How about the dash lights?

My thinking is either the switch is bad or you are getting a power feedback from a different system.

To see if it is the switch you could unplug its connector. Normally I would have you just pull the fuse but in this case that fuse also powers two of the items that could be feeding power to the lights. Disconnecting the switch itself should show if it is the problem.

To remove the switch you need to remove the trim over it, then remove the knob by pushing the retaining pin. Then unscrew the nut and disconnect the switch.
If the lights go out it is the switch.

Was afraid of that. You are getting a power feedback. Likely from the tow harness. Go to the under hood fuse box and pull fuses D, F, G, Q and T one at a time and see if one of them turns out the lights. They are the towing harness fuses and the brake controller feeds.
If the tow harness was installed like they typically are it is possible the wiring is shorted together. It would also explain the tail lights flickering.