This seems to make it clear that the shift interlock is releasing. Being able to move it just enough to get the reverse lights suggests that something is jammed but perhaps the cable has enough give in it to allow a little bit of motion.
I wouldn’t expect that backing into a snow bank would get snow packed around the shift linkage, as that’s probably a couple feet off the ground and in the front, but depending on the particular situation it could be possible. If time allows, it makes sense to see if it works okay after it thaws.
One approach would be to disconnect the shift cable from the shift lever on the transmission. You could then see if the shifter (inside the car) moves normally, and see if the lever on the transmission can be moved manually (make sure the car can’t roll when doing this). If the parking pawl is bound by the weight of the vehicle, jacking up a front wheel off the ground should alleviate it.
Usually there’s a setscrew to allow the handle to be removed. Unfortunately, my repair info has virtually nothing on the transmission and shifter, assuming it’s not a CVT.
Let me know the engine size in liters, and whether or not it’s a CVT, and I may be able to come up with more info.