While you can't fish at Cleveland Park, there are several other locations in Spartanburg County where you are allowed to fish:
Lake Blalock
Lake Bowen
Lake Cooley
Lake Lyman
Tyger River
Holston Creek Park
Lake Edwin Johnson
Lake Tom Moore Craig
Click here for more information about each location, including how to find the fishing spots at each one: https://www.dnr.sc.gov/lakes/access/spartanburg.html