Reverse lights on in Park gear?

There is only one item that can cause this issue which is the park/neutral switch on the left side of the transmission. Is this what you replaced? Here are the reverse light wiring diagrams in the images below and how to correctly replace the switch as well. Please let me know.

Yes, that is the part I replaced, the park/neutral switch and the reverse light still stays on in Park.

There might be some wiring problems, as you can see from the wiring diagrams there is not much to the reverse light system. Can you remove the rear light housing to look for wiring problems? Also remove the bulbs to see if something has melted.

Both bulbs are good, no melting around any sockets in either of the taillights.
I’ve pulled back some of the wire sheathing and the wires look good, no rubbing, pinched or exposed sections. How could I further check for wiring problems?

Does the vehicle have trailer wiring? Also, please check out the reverse wiring diagram above because there is really nothing that can cause this problem except the park neutral switch.