Love all your signs, pictures, and stat boards. You're right! (Within the USA) the FAA determines the control of airspace. The FAA also makes it ever so easy for any person or entity to make the area under their control a NFZ, schools, businesses, residential,. Which in the past I have read stories of so called drone pilots making the homes as NFZ to keep out other possible intruders. Funny thing is, if they fly in their own yard, they violate the NFZ as well. The OP on this issue, I believe, most likely made efforts to determine before hand as to permissible flying or not, often signage is sporadically posted and often times well inside the park. Yes, the flight takeoff and landing must occur outside the park to be legit.
I have another question to ponder . . . If I place my drone outside the park and takeoff, then fly said drone into the airspace over the park, is it acceptable for me to come into the said park while maintaining flight control of the drone? . . . or must I remain outside the park?