Moving to another prefecture with a car

In general
– I have to notify my current city office, then notify the new city office after moving
– Update zairyu and my number cards
– Update addresses in post office, bank, credit card company.

But what is the procedure for the car? Do I have to register it again in the new city? Last time all the registration and stuff was done by the dealership when I bought it. I'm assuming I need to get a new parking certificate as well?

Let me know if I'm missing something (car related or even in general)

Thank you!