To ask who I’m supposed to report my CF neighbour to?

Not worth the agro to get involved with using bolt cutters yourself. Just get the local council involved.
Why not throw a community picnic in the park every weekend?? All invited!! Publicise it with a "bring your own food and drink", ask people with kids to bring stuff for the kids to play with (soccer balls, cricket set etc etc) - that'll really wind up your obnoxious neighbour :) No doubt there are lots of people in your neighbourhood who would love some community get-togethers that make use of the lovely park area - everything from "community cook-outs" (or "potlucks" as my friends in the states call them - I loved those when I lived there - all the neighbours and friends would get together in the local park and people would each bring a dish or two (you'd usually have a little sign up sheet so people would say they'd bring a starter/side/dessert or drinks or whatnot - just so you didn't end up with 25 bowls of potato salad lol!); people would bring a load of different sports equipment/party stuff that was easy for all to play (Cornhole boards, boules, horseshoes etc etc - so many different things these days!) and it would be a good time had by all.

You could also set up a community yoga, or running club, or meditation etc etc. Tons of different ways to use the park. All of which will no doubt drive your nasty neighbour bonkers. But they should have thought about that before they decided to buy a house that's right next to a park!

@BarkItOff this thread I just found has some useful information in it, but you'll need to sift through to see the parts which may apply to your situation:

https://www.mumsnet.com/talk/property/5044485-would-you-buy-a-house-on-a-private-road

Ok. The developers will have been the original freeholder. As and when houses were sold on your development, the freehold of the land on which each house sits will have been transferred to the home owner.

However, the freehold of any shared spaces - in this case your unadopted private road - can only be owned by one organisation. This will have originally remained the property of the developer, hence why they appointed a management company, but it is quite normal for a developer to sell it off when they wish to walk away from it.

Typically, when they wish to sell, residents who use the shared areas have to be given first refusal to buy it, which they would do by getting together and forming as limited company (it is more complicated than this, but that's the nuts & bolts of it). When this is done, each homeowner owns a % share of the shared area, and pays towards the up keep of it. But if they don't buy it, another company can.

So, the road is either still owned by the developer, or it's owned by everyone on the estate because the developer sold it to them and they formed a company, or it's owned by another company altogether, but no matter who owns it, there has to be a limited company in control of it - and if you don't know who that is then the management company should darn well be telling you.

The best person you can speak to is the person who's lived on the development the longest, from new, as they should know all about what's gone on. I can only conclude you must have moved in some time afterwards, because had you been there from the outset then you would have been made aware that the developer had sold off the freehold of the road. Did your solicitor not tell you any of this when you bought the place?

Have you considered holding a festival on the park opposite her house?

So a very strongly worded letter to the management company telling them to get their finger out of their arse and sort it and another approach to the local authority telling them the same oh and raise it with your local county councillor, they can get stuff done. (I'm a former parish councillor).
If you really want to you can get a group of residents to sign a letter/petition to the CF neighbour telling her to remove the lock or you will take further action.

It's perfectly legal to cut a padlock that has been illegally installed. Even if the park is private it is not her property, it belongs equally to all residents of the estate and the dogwalkers wishes to have the gate open is just as valid as her wish to have it shut, she is not the boss.

Bolt cutters

I’d instruct a solicitor and get them to write to him with all the necessary evidence to remind him. A solicitor will do this (I work for a law firm). It might make him stop.

In that case, are you quite sure the developers are no longer involved? Because they will have been the original freeholders and will have released the freehold to each property as and when it was sold. However, the shared area (in your case a road) can only belong to one organisation, and if you are the freeholder of it that means you own your % share of that freehold and are part of the company that was set up to administer it.

So in other words, it either belongs to the developer, or it was sold off to the residents who will have to have formed a limited company together in order to own a % share of it.

Who maintains the park, ie. Who cuts the grass, clears rubbish, empties the bins, carries out safety checks etc? It is their responsibility to ensure that access to the park is just that. Accessible. It doesn't matter that there is another gate.
I assume that this is either a parish or town council or the local authority and they will be able to enforce the removal of the padlock. If it is the management company then despite them denying it, it is in fact their legal responsibility and they must remove the padlock.

So in that case does it mean any Tom, Dick, & Harry can come along & chain them? Does it also tell you who the directors are, or at the very lease the name of the limited company set up in order to allow you a share of the freehold of the road?

By lease I am mean the rule book if you like, that state what you can and can't do on areas of a shared freehold (for example, chaining off gates).

You are a freeholder, but that will that you own your % share of the freehold. That's why you have a lease, which ensures that everyone knows how to behave.

So as I said in my PP, if you are the freeholder of the land on which your house sits, and a % freeholder of the road & the associated park space, there has to be a limited company and directors of it.

If they are not serving you well, the management company appointed by the developers can be removed and replaced by the directors of the freeholders, this is perfectly normal in, say, a block of purpose built flats, where management companies get brought in and out (I know this as I carry out work on sites where this happens, and every so often I am told that I will be sent instructions from a different company and need to change who I invoice accordingly).

Whilst it may be the case that what's happened is outside of the scope of the management company (and for what it's worth I think that's a right cop out), they should at the very least have signposted you to the directors of the company that has been set up to administer your share of the freehold.

Can you do a diagram?

Ok, so this is moving in the right direction now. You are freeholders who own your % share. This means a limited company must exist, and there will be directors of it. The directors are the ones you need to raise it with, and you should all have a lease which explains all the rules & regs of the land you are a part freeholder of.

Ok, so told by who, and where is the documentation to support this? Which residents were responsible for appointing the management company who is refusing to help you? Because collectively you can hoof them out and appoint another, if you are a % owner of the road. Did the solicitor not go through all of this in detail when you bought the place?

Also as a PP mentions, why are you paying towards the upkeep of a public park?