"...looks like it’s been there for at least the 12 years required"
Looks aren't enough though. You need evidence. This could be by way of a statement from the previous owner that they had lived in the property for x years and the fence had been in that position all that time for example.
However, the really big thing is to find out whether the park is registered land or not. Nowadays, just about all homes are registered with HM Land Registry but this doesn't apply to all Local Authority land, so it may be that the park is unregistered.
However, there has been a big push to get all Local Authority registered over the last few years and the plan is to get it all registered by 2025.
You can check to see if the park is registered in a couple of places for free either here:
https://www.landregistry-uk.com/map-search
or here;
https://search-property-information.service.gov.uk/search/map-search/find-by-map
For example, in the city where I live, both the park in the city and the one on the edge of the city have both been registered.
If the park has been registered then I really wouldn't bother. In that case HM Land Registry would then write to the Local Authority telling them that beachguy wants to take part of the park and asking if they're ok with that.
They will object and it is then down to a court case. You would need to speak to a solicitor in that situation.
In this case, if you reasonably believed that the fence was in the correct position and it had been there for ten years then you would have a strong case.
If it had been you that moved the fence then that obviously wouldn't apply, but if a previous owner had done it that would be a different matter.
Is the extra bit of garden worth going to court over?