If the museum a few blocks away is open, then you can park in the lot behind the building. And if you go down to the museum, they will give you a free parking token. Then you can browse the whole area.
There is a public parking garage about 2 blocks north in Kleman Plaza. Last time I was there you could park for the day for about five bucks.
Visitor parking in the Capitol area is limited to either metered street spaces or designated lots. The City of Tallahassee manages two parking garages (Kleman Plaza and Eastside Parking Garage) and one surface lot (near Duval and Gaines). The Department of Management Services (DMS) manages two lots (Lot E and Lot 4). The dolphin fountain is at the front of the Capitol. It is a drop off spot only.
Thanks! Your answer is awaiting moderation.