I would definitely call whoever is in charge of the parks in your area to find out. You may not need one but the people in charge may be concerned about liability and may want to have proof of your liability insurance. Whether a permit is necessary may depend on the location. I am sure that cities handle that differently.
Even if this adds another layer of complexity to your plans, you need to stay on the high road here. In fact, it may even give you an opening to talk to people in your city’s parks and recreation program. They may be delighted to know of you.
I wish you good luck.