There is plenty of parking at the Miami Swim with Dolphin Tours and Ticket location. It cost $10 per car to park. Of course there are many handicapped and maternity parking spaces too.
That price of $10 is for all day parking. The only time parking is very limited is during Christmas and Easter Week. To help the attendant at the gate, please make sure you bring cash to pay the $10. They do take credit cards at the gate now but this definitely slows down the line of cars that patiently waiting to get into the park.
The front gate is in the middle of the park, so you can park to the left or the right of the main entrance. If you park to the right, you would be able to walk to the beach of Key Biscayne. There is a walking path from the park to the beach.