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Hello, if the shift linkage is moving as normal then, It has what is called a "parking pawl" inside the transmission that is supposed to lock into a slot to prevent the final drive from engaging and there is a fault with it. In most cases its because it has broken and the transmission has to be removed to replace it.
Thats a good sign that the pawl isnt actually broken though!
The interlock switch only prevents the shifter from being able to move out of park unless the key is on and the brake pedal is depressed so this is in a different area. Same with the neutral switch. Those are both electrical controled devices and the issue you are having is going to be more mechanical. Probably something in the actual linkage but the good news is that the trans wont need to come out OR have to be rebuilt!
It will have to be a hands on inspection at a shop to hunt it down.