https://freecampsites.net tends to be pretty good about listing county parks but not always. Finding county parks tend to be iffy. Also there are some County parks that used to be State Parks. I know of a few that have been transferred. They tend to be small, little-known parks.
Your best bet for finding a county park is google.com/maps
Type into the search bar at the top of the map "Campgrounds near (city or county, State)" This search will pull up both private and public campgrounds. You will have to sort thru them yourself. I suggest that as you run across them, you make your own "directory". I like to build spreadsheets and have made a "Campground Log" for listing campgrounds that I would like to visit or have visited in them. Several are small public parks and I try to add as much info for them as possible. When my husband & I first started tent camping, I used to keep all the info in a small notebook. With computers, I have shifted over to electronic and update the info every so often because everything changes.
You might like this one, it's a good example of a former state park now county/city park:
Sometimes listed as a state park, Harry McAdams in Hobbs NM is now a county or city park that is backed up to a public golf https://www.campendium.com/harry-mcadams-campground