A little known park in Tampa is the Cypress Point Park. It's located where West Cypress Street dead-ends before you run into Tampa Bay. Turn left at the dead-end and follow the road into the park.
It doesn't have any playgrounds for the kids, but it's a great place to have a picnic. There are several pavilions that you can probably reserve or just take one if no one is there. They have grills there to use as well, and a sign said that you can't take your own grill. Which is just as well, because where will you dump the hot coals and who wants a smoke smelling grill in your car anyways?
It does have a real nice beach overlooking Tampa Bay, public restrooms, drinking fountains and a real nice beach volleyball court out by the parking lot. Dogs must be kept on a leash but no dogs are allowed on the beach.
Check out the photos below by clicking them to make them larger. Here's a link with more info below.
http://www.tampagov.net/parks_search_webapp/ParkDetail.aspx?nbr=28