Car parking in private road that I live in

Hello,
For context, I live in a small cul-de-sac that’s a gated community private road with less than 20 houses (in England).
The houses are mostly 4/5 bedroom houses but nearly all of them only have 2 car driveways.

In my house there are me, my wife and 2 grown up children, all who drive and own cars. We park 2 on our drive and one in a communal parking spot (when available).

Parking is always very considerate (we leave a minimum of 3.1m between car and the kerb opposite so dustbin lorries/emergency vehicles can get through). Our neighbour is in a similar predicament but also parks considerately, as is another neighbour a few doors down.

The problem we have is an ‘echo chamber’ of the car parking being a ‘health & safety’ issue, despite everything I’ve already mentioned. We have an estate management company and liability insurance. However, after stopping all parking in the entrance to our road, and one resident in particular being constantly vocal about parking in the road, he has now called a directors meeting (I am one of the directors) on the back of a pack of lies, where his objective is to prevent residents parking outside their houses. If successful, this would mean every car not on a drive and every guest will have to park in the next (public) road, causing a nuisance to all the residents there.

My question is, in a private road, can any laws be enforced to prevent a resident parking in front of their own house, where it can be proven no obstruction is taking place.

As a footnote, the resident that’s really pushing this has recently sold his house (he did send private messages to residents that park in the road asking us not to park in the road when he was having viewings so it looks better, which to me sounds like misrepresentation). Should this be declared on his TA6 as an unresolved dispute, being as he’s raised it with the estate management company?

This has started to become almost like high level harassment, weaponising the management company against us by sending letters to everyone in the road. I am speaking to the management company concerned.

TLDR;
Can parking rules be legally enforced in a private road to stop residents parking in front of their own houses when no parking laws are being broken?