I've not read the full thread but I am a keen parkrunner and go most weeks (not to Little Stoke). The ethos of parkrun is that it is free for all. Very few people earn money out of parkrun other than a few staff. Equipment is paid for by sponsorship.
So if a charge is going to be levied, parkrun does not have cash to pay. There is no money. It's that simple. It is not like a football club which levies subs, so it has some income from which it can eg pay for pitch hire. Many councils charge businesses - eg if you are a personal trainer. I'm not sure I even agree with that as everyone pays council tax.
The people who "staff" the events are all volunteers and cannot be asked to collect money. If the council paid someone to collect the money it would defeat the object of collecting it. And if people just turn up and run in the park, they can't charge them because anyone can run in a park. So really it's not enforceable. However, the terms of parkrun's insurance dictate that consent for the event is given. If the council withdraws consent, that means no insurance so no event.
The council concerned had other options. It could have charged for parking (many venues do, this is not an issue as you can car share, get the bus, run, cycle etc if you don't want to pay to park). However, its "parent" council South Gloucester has a policy of no charging. Maybe they should have been more flexible about that.
The council also said that people from outside the area use parkrun. Yes and people from Bristol do parkruns elsewhere. Councils often say things like this - eg you can't use our library or rubbish tip because you live over the border - very silly and there should be an assumption of reciprocity rather than erecting barriers. There is no parkrun in my district but plenty within a 30 minute radius. But if all councils took this view I wouldn't be able to take part in one.
The council also had developer funding from local new housing developments but chose to use it for football faciliites. And it has healthy cash reserves.
It wasn't to do with money.
My guess is that there are simply too many runners now (it used be 60, now there are around 300) and someone somewhere had their nose put out of joint (eg dog walker who's married to a councillor and didn't like sharing the park with runners at 9am, even though for the rest of the day they wouldn't have to). Read the minutes of the meetings and you'll see a few residents' comments to the effect that the park is not just for runners. Parkrun is at 9am for up to an hour. That's all. Junior parkrun is shorter and takes place on Sunday morning. But if they simply wanted parkrun gone it's a pity they didn't just say that. I suspect "we can't cope with 300 runners, you've got too big for the park and we need you to find a different venue" would have caused a lot less adverse publicity than "pay us to use the park".