Regarding local power to regulate drones, see City of Burbank v. Lockheed Air Terminal, 411 U.S. 624: they can, but the regulations are subject to strict scrutiny. That means that it has to be designed for a compelling government interest (such as safety), and has to be the least restrictive way to do that. If their "bottom line" statement is accurate with respect to the ordinance, this also means that you cannot operate radio-controlled toy car in those spaces, either. It may be that the Open Space land is analogous to a wildlife preserve, so their compelling interest would be avoiding "impact" on the park. The regulation does not appear to be generally about all parks. You could, I suppose, throw a drone from place to place inside the park on the reasonable presumption that that doesn't constitute "operating".
Since the purposed regulation prohibiting drones is not published where I can find it, it is hard to know what they claim to restrict. They would have the right to prohibit flying a drone over the Open Space zone, regardless of where you are standing. Supposing the wording were something like "it is prohibited to operate a drone in the park", there is an ambiguity: does that refer to the location of the operator or of the drone. Supposing you sued them because they ticketed you because your drone was over the part but you yourself were outside the part, I have a hard time imaging an appeals court saying that it's okay to fly over as long as the operator is outside. It would come down to some assertion about the "plain meaning" of the regulation, supported by testimony about why the made the regulation (it's about drones being in the part, not drone operators).
A regulation about skipping on your left foot, unless you're wearing sandals, would surely fail strict scrutiny.
There is a further possibility that US law does not allow non-federal regulation of space above the ground. For example, here they assert that there cannot be additional local registration requirements, and that is probably correct given existing federal law. This FAA document suggests (does not mandate) consulation with FAA over proposed rules which impose
restrictions on flight altitude, flight paths; operational bans; any
regulation of the navigable airspace. For example – a city ordinance
banning anyone from operating UAS within the city limits, within the
air space of the city, or within certain distances of landmarks.
This falls short of asserting a local lack of authority. Whether or not FAA can prohibit municipalities from declaring no-fly zones is a matter that will probably be decided once a city files a lawsuit.