You will want to check with your city and county ordinances to find that out. Different places have different requirements. Typically, you have to have a license in order to conduct a business in a public setting, or you have to be a member of the parks department. For example, I work for a parks and rec department that provides outdoor classes, and anyone who teaches outdoors is an employee of that parks and rec. I couldn’t randomly go onto their land and start charging $ to teach.
This video from the IDEA site on getting permits for outdoor fitness might also be informative to you.