How Would You Transfer A License Plate In The State Of Florida?

Okay… the situation is that my parent cosigned for my car and the loan to pay the car iff. Its a Hyundai Elantra 04. My parent said I would be responsible for paying the fees for the title transfer. I went down to the DMV about a week ago and they said it be 400$ with title transfer and new registration.
The car is registered and paid by me. Does my parent have to transfer title and tag out of his name and over to me? Is this possible?
I’ve looked at the website nearly a dozen times and still cannot get answers. Why should I pay 400$ on a car when my parent is still sole owner of it by default/cosigner on loan??