Actually, the Division of Recreation and Parks does have the authority to ban firearms from state parks under FSS 258.007, their rule making authority. It is specifically prohibited in fac62d-2 [ 62D-2.014 Activities and Recreation. section 10] [link below]:

(10) Hunting and Firearms. Hunting, trapping or the pursuit of wildlife is prohibited on all park property, except in Reserves,
as authorized by the Florida Fish and Wildlife Conservation Commission. No person shall use, carry, or possess in any park
weapons such as firearms of any type, air rifles, spring guns, bows and arrows, gigs (except in areas where gigs may be legally used
for saltwater fishing), sling shots, or any other forms of weapons or trapping devices potentially dangerous to wildlife or human
safety except when such weapons or traps are used for resource management purposes as authorized in this subsection.Shooting
into park areas from beyond park boundaries is prohibited. Any device which is employed to kill, immobilize, or capture any
wildlife or any device otherwise used in violation of this chapter shall be seized and confiscated by law enforcement officers. The
Division may authorize the control of nuisance animals and may remove all exotic animals from parks by trapping and other
necessary means for park resource management purposes. Such authorization shall be in the form of a license, permit, or contract
negotiated by the parties or made pursuant to an advertised bid by the Division. Authorization may be obtained in the same manner
as stated in subsection 62D-2.013(7), F.A.C., herein.

http://www.dep.state.fl.us/Law/docum...k/fac62d-2.pdf

FSS 790 specifically reserves the right to control firearms to the State of Florida, of which DEP is a part. And the state legislature has empowered them to make such rules as are necessary to manage state parks. Unlike counties and cities, possession of firearms in a state park not only can be restricted, it has been. And it carries the weight of law.

Check me on this, please. But I think that I would respect the no firearms signs at state parks.