If you rent a scooter while staying at the dolphin, how do you transfer it to the parks?

There are routes available for the park.

You can walk/scooter or take a otter friendly boats and busses

The dolphin no longer has Disney busses that come directly. We found it easier to take the little cruise boats to Hollywood studios or Epcot, or you can walk/scoot to the Yacht club resort and from there you can get all Disney busses.

The buses and boats accommodate scooters

All transportation system will accommodate the scooters, the drivers assist you on and off the vehicle and make sure it is secured.

Not sure if scooters are allowed on skyliner or boats, but definitely on buses. You ride it to the bus queue and the bus driver will assist you onboard and secure the scooter for transit.

Bus routes are available to the parks from Dolphin and Swan. The scooter user should be able to board the bus, and offload at the parks. Call to verify, though.