Since being laid off from the space program, I haven't been back to the Kennedy Space Center Employee Rec Area/Campground (KARS 1) but it was nice, cheap, pet-friendly and river-waterfront!!! Next best park IMO is Jetty Park as mentioned above.
Here's a link with details on who can access this NASA park: Kennedy Space Center NASA Exchange Council - KARS Park
I know some folks enjoy Manatee Hammock in Titusville (also river-waterfront).Manatee Hammock Park Campground Home
Now that I look at this link, I remember Longpoint as a kid but don't know the condition now. And Wickham park is okay - and it has a nice dog park.
Good luck - I miss "home".