It may be laughable but it is true, although I dunno about 'dramatically'. Obviously the whole place isn't going to fall apart in a month but if you actually think hundreds of people pounding round the paths of a park have no effect whatsoever compared to the normal traffic levels (and leisurely pace people travel at), you're a fool. Plus you know what people are like, there'll be piles of discarded energy gel bags and bar wrappers and water bottles and god knows what to clear up afterwards.
Now, I happen to agree with you that we all pay for the amenities so we should all get to use them as and how we see fit and if they cost a bit more to maintain I wouldn't mind paying, I dunno, a tenner a year's more council tax. But there are several 'park run' organisation and while the main one is non-profit, not all of them are. And if someone is charging each person to run round a public park (which they could run around for free anyway), simply for the privilege of having their time logged, and making a nice bit of profit off it I think that money or at least a chunk of it should go to the people who have to run the venue where it happens.
Also, unless you live in Manchester (which is effectively the pilot for a new arrangement) any health cost reductions due to the participants getting fitter don't reduce the council bills one penny, because the council doesn't have anything to do with things that are I'm[proved with it. Yes, they may end up with a 1% smaller care bill in 20 years time but believe me, the way things are going there won't be any council care in 20 years time. The councils need money NOW, and most of them aren't particularly choosey about how they get it because they're using it to stop people dying right now, not in 20 years time..