No National parks that have beaches, but there is a State park: Ft. De Soto State Park, just south of St. Petersburg.
Fort De Soto Park is a Pinellas county park. It is a great place to visit and includes a Spanish American War era fort, a couple of fishing piers, picnic shelters and long stretches of beach with plenty of parking.
(notice pinellascounty.org in the web address)
Here is a list of National Parks in Florida.
Although some of the parks listed are beaches they are not near Redington Beach.
Redington Beach is one of several beach communities that are on the Gulf between Clearwater Beach on the north and St. Pete Beach on the south.
An overview of Pinellas County Beach Communities
Pinellas County has many wonderful beach towns; some are well known and others less well known, but the beaches are essentially the same because each community runs into the next, so for people who are traveling to the area here is an overview of the beaches.
Beginning at the north end is Clearwater Beach on the aptly named Clearwater Beach Island. Clearwater Beach is a part of the City of Clearwater.
Crossing the Clearwater Pass Bridge southward is Sand Key. Sand Key is about 14 miles long with the Clearwater Pass at the north end of the island and John's Pass at the south end. Sand Key includes several communities: Sand Key, Belleair Beach, Belleair Shores, Indian Rocks Beach, Indian Shores, Redington Shores, North Redington Beach, Redington Beach, and Madeira Beach.
Treasure Island is separated on the north by John's Pass and Blind Pass on the south. The southern end of the island is known as Sunset Beach.
South of Treasure Island is Long Key which includes St. Pete Beach and the southern tip of the island is Pass-a-Grille Beach.
Hope this helps.