Yes, but they would need to pursue a case through the civil courts which is going to be expensive for both parties. The article also isn't clear whether this is a byelaw or just a "policy" (which is how it is described in the article, whatever that means), if it's a formally set byelaw that is clearly stated on the park signage then I suspect they'll have more chance of success in court than if it's just a "policy" that is not clearly communicated.
Realistically, all they (or any other council) need to do is enforce the Drone Code and the various applicable nuisance laws, and maybe have their park workers, community officers, etc. perform a spot check of any pilots they come across to ensure they are in compliance with CAA's new registration scheme. For particularly busy parks (which these may well be) that alone may be enough to preclude any legal flights and, unlike the local rules the council can pass, those would be an actual criminal offence.
Another reason for this being dumb: once commercial drone deliveries etc. start (which they eventually will), which is going to be safer in the event of a drone falling out of the sky:
Option A: flying around the edge of the park, presumably following a public road and/or footpath route?
Option B: flying over the park, which is mostly going to be open space, and probably dotted with trees to potentially break a drone's fall before it hits someone?