I think you would be wasting your time but there is no harm in asking. Suggest try talking to the Highways Department - try for a meeting where you can show them the title plan.
As far as I know, the requirement of a Local Authority is to maintain a list of public roads. There is a separate register of public rights of way (that are not roads) and another list of Register of Commons, Common land and Village Greens. Any definitive maps tied to these lists is unlikely to show the clarity of detail you require. Also, as far as I know, none of the above categories of land use are tied to the Land Registry in terms of land title (in other words, neither the Local Authority or anyone else 'owns' them as such).
So if you have a land title plan showing that you own a piece of the (current) public highway, that may be interesting.
The business about a piece of land ending up being a public right of way is a different story. There is a principle of unchallenged use of a piece of land by several people that can lead to it acquiring a public right of way status. This of course these days largely applies to public footpaths or bridleways, where the land underneath the path belongs to someone. The historic principle is of course how roads came to be in existence in this country, long before the need to protect one's scrap of land by legal means. All I am saying is that after 20 years, you may find you have forfeited to the piece of land into a public road.
See this for a better explanation of the principle.
http://www.solihull.gov.uk/transport/publicrig htsofway_14232.htm
If it does end up that the Highways Dept say the land is now part of the public highway, you could try and persuade them to put parking restrictions on it - in the form of yellow lines.