You'll need to discuss this with the council, assuming they actually own it - they may not be keen to sell it but may well be prepared to grant a right of access.

It is highly unlikely you will be able to purchase it. It is legally part of the public highway.

The person you purchased it from could 'write down' absolutely anything they liked. If it isn't in the deeds, then you have no legal rights at all.

Having said all that, if the piece of land is completely useless to anyone else but yourself, you could always ask the council about purchasing it. However, even if they decided to sell it to you, you'll have to pay both sets of legal fees, along with the purchase price, and you'd almost certainly have to take out a full indemnity policy as well.

Alternatively, just keep on parking there like you already are, and don't stir anything up.

My limited experience with trying to purchase land adjacent to my property - privately owned, not by the council - is that potential sellers tend to think in terms of sometimes far-fetched ideas of what use the land might be to them in the future, and refuse to sell.

In your case, the council might believe there is some remote possibility that the land might be useful in case of redevelopment, to provide access or whatever.

On the other hand, a good offer might sway them. I suggest you approach the council informally, say by getting a few minutes with a planning officer, and put out some feelers.

However, even if you're successful you might be parting with a lot of money with little benefit to yourself. As someone has already said, you could probably just park there without any objection.