No, you cannot have your car registered in Florida and have New York insurance on it. When your car is registered in Florida the states requires you to have auto insurance that is issued through a Florida agent with an insurance company licensed to sell in Florida.
So Florida requires that your insurance be from a Florida licensed agent when your car is registered in this state, just as New York requires insurance from a NYS authorized insurance agency when your car is registered there.
If you are moving in with your parents in New York and becoming a New York State resident then you are required to register, plate and insure your car in NYS.
The New York State Department of Motor Vehicles states that you must apply for a NYS driver license not more than 30 days after you become a NYS resident. You also have 30 days after you become a resident to register your vehicle in NYS.
When you apply for a New York title and registration you must have valid NYS liability insurance. You cannot use proof of insurance from another state under any circumstances according to the NYS DMV. Out of state insurance coverage is not acceptable or valid to register a vehicle in New York.
To get car insurance prices for New York State, click here. If you register your car in NYS and are living with your parents though you should be able to have you and your car added to your parent's insurance policy.