I'll be honest with you - Gainesville isn't the best place for parks. However, just outside Gainesville, you have a plethora of choices.
The Florida Natural History Museum, on UF campus, has an extensive butterfly rainforest, and some wonderful other exhibits on local flora and fauna.
The Harn Art museum, also on UF campus, has rotating exhibits ranging from modern art to ancient beadworking.
Fred Bear Museum, a showcase of big game collected from around the world by the late Fred Bear, expert bow hunter and founder of Bear Archery.
The Devil's Milhopper is a local geological sinkhole with steps all the way down, and so filled with greenery it's like descending into a rain forest.
The Ginnie Springs/Poe Springs area has crystal clear springs, and a great park to camp, play, and swim in.
Kanapaha Botanical Gardens is a beautiful setting, both sculptured and wild, nestled around a lake, with lovely flowers of all kinds growing.
Payne's Prairie, just south of Gainesville, is a 21,000 savanna with bison, alligators, and over 250 species of birds. Hiking, horseback riding and canoeing is available.
Kanapaha Veteran's Memorial park is on Tower Road, and offers woods, rolling hills and places for cookouts and sports such as field hockey, soccer.