It depends on the Park District. In Chicago you need to have a permit to use any of the parks. You need to call your Park District and ask, they may even require you to have liability insurance before they consider to let you use any park. You may get away with out anyone finding out that you are using their park, but if they do they may ask you to leave and don’t go back until you have gone thru them. Good luck!
I have a business license and am allowed to use my local parks.
Every city has it’s own policies, I suggest finding out the requirements prior to starting your bootcamp as it would be really humiliating to be asked to leave in the middle of a workout.!
Liability Insurance for outside training is a good idea along with waivers that are detailed towards outdoor training, as there are episodes that can occur that you may not be aware of such as uneven pavement obstacles in your way that a bootcamper can trip over.
We use a small park and are very respectful of the neighbors and others using it.