Like many towns, I think Maplewood has a "no overnight street parking" rule -- period. There are no permits, and few exemptions. I guess it's because most houses have driveways.
You do need to register your car in NJ if you're moving to NJ, but that's a separate issue, it's a state law not a Maplewood law.
There is a parking permit for downtown Maplewood (near the train station) for residents only if you want to drive and park near the train station. It wasn't always like that, but I guess it got more crowded over the years, so now you need to pay to park on the streets near the train station during the day (or you can pay in the lot).
You can get a lot of questions answered here:
Maplewood — Maplewood Township Committee