So are there two separate dog parks between this location and the "Fort De Soto park" south of here?

Dog park nearest to gulf fishing pier has great views.

There is a big dog and a small dog park at fort de soto i don't know if that helps you or not

Yes.

There is one designated dog park on Fort Desoto , with two sides for different sized dogs. Dogs are allowed on some stretches of beach in Ft Desoto just fyi