Picnic tables, shelters Playground Historic features; Gardens; Hiking trails; Bicycling trails: Paved Our new, state-of-the-art facility offers the best possible care for your pet. This 126-acre park along the Reedy River features tennis courts, a sand volleyball court, half basketball courts, a softball field, playgrounds, a fitness trail, and picnic shelters. It was built with WPA funds, as was the adjacent skating rink. Cleveland is an unincorporated community in Greenville County, South Carolina, United States.Cleveland is located on U.S. Route 276 and South Carolina Highway 11, 8.7 miles (14.0 km) north-northwest of Travelers Rest.Cleveland has a post office with ZIP code 29635, which opened on … See all 143 photos taken at Cleveland Park by 1,962 visitors. Hours: Dawn to dusk. Turns out that my meeting wasnt as far as, I visited with my 3 children 3 and under today while on vacation in the area. There is a nice stream that runs through the park. She joined the Cleveland Park family in 2014, and has enjoyed every minute of it. 10 reviews of Cleveland Park Animal Hospital "Dr. Finley and staff did a tremendous job caring for Tillman the past 5.5 years... today they saw me early in the morning and did some xrays and referred us to Upstate Veterinary Specialists for back surgery due to an injury. nice if it had a dog park area. It is also home to the Greenville Zoo.