It's not uncommon, in rural areas, for a property owner to own the land right up to the centre line of the roadway (with the owner on the other side owning the land on the other side of the centre line). The fact that the planning regulations require them to set the boundary wall/fence some way back doesn't change the fact that they own the land right up to the centre line.
But, in relation to the strip of land which has been dedicated as a public road, their rights of ownership are subordinated to the public's right to use the land as a road.
Members of the public have a right to pass along the roadway, with or without vehicles. The owners cannot prevent this.
The question here is whether the rights of members of the public extend to parking on land dedicated as a public road. Believe it or not, there isn't an absolutely authoritative ruling about this, so far as I know. But the general view is that parking your vehicle, as an incident of using the roadway, is covered (subject to any road traffic regulations which may forbid parking, or impose a charge for parking, in a particular location).
So, parking your car on the road beside, or reasonably convenient to, the property you are visiting, for the duration of your visit, is probably fine, so long as it's not parked in a way that causes a nuisance (blocking a gate, impeding motorists' line of sight, that kind of thing). But parking your car as a form of indefinite storage, until you next happen to need it, would not be.