Don't recall any adjustment on the switch? But I will check later.
Have you got a standard lever operated shifter or a cable operated aftermarket shifter?
The cable ones often do not work right. Mine has the same problem as yours no matter how I adjust the cable. I suspect the lever on the selector switch is not exactly the right length. I intend to look into this and correct it over this winter.
If you have a standard set-up have you got any bent levers or worn pivot joints?
No but one of the levers could be bent making it just short enough to not quite click into Park correctly? I forgot to check but I am pretty sure there is no adjustment at the trans end. I have no experience with column shifted Challengers. They are pretty rare.
I have looked into this now. Below is what the manual says. There is in fact an adjustment on the actual linkage.
You could also try removing the switch from the trans (catch the fluid in a container) and see if the switch operating fingers are centered in the switch opening in the case when you have it in park.
Check your neutral safety switch.
Make sure that it is ok, and check and make sure you have power at the connector.
Not an uncommon problem with any and all makes of automatic transmissions.