They're all good in the Pan Handle. If you get a state park on the beach ... like Port St Joe or St. Andrews, pets are welcome, but not on the beaches. You'll need to find other areas if you want to take your dog on the beach. Inland State Parks don't have that problem of course.
And if you decide to stay at Florida Caverns State Park (which I highly recommend), if it rains in Georgia, you'll get flooded out in the park 1 week later! Don't ask how I know.