There's no specific legislation to prevent it. Just as when parking on any other private land, the owner of that land/road can charge for parking there but only when there are clearly displayed signs to that effect and where those charges aren't wholly disproportionate.
For example, there's a business park in Ipswich (with private roads throughout) where there are clear signs that parking on any roadway isn't permitted, with a penalty charge being made for doing so. There are frequent patrols of those roads, with drivers being given penalty charge notices for the smallest infringements of the rules.
However your right (or lack of it, depending upon your view of things) to park on a private residential road is exactly the same as that as a total stranger's right (or lack of it, as above) to park on the driveway of your house. It's not illegal but it might well cause a mighty big argument!