I wouldn't be too sure of that. It's more likely that the parking space was sold as part of the property. Only the brown shaded area is common and subject to your rights of access.
Dunno what he's trying to say here. When the original owner died, the title would have passed to his heirs - either as directed in his will, or by the rules applied if he dies intestate.
Not in this case. You are thinking of what is often called "squatters rights", or more correctly (I believe) as possessive title. To gain possessive title (briefly) you need to have occupied the land for 10 years - without permission. In this case, the possession was "with permission" as so doesn't count.
I did hear of a case locally a few years ago. Someone bought a house that came with a lovely big garden. After purchase he got a letter along the lines of "The land <description> is mine, you can continue to use it but it remains mine and I reserve the right to ask for it back at any time". A good chunk of the garden wasn't actually the vendor's to sell and was only used by permission of the actual owner - something which apparently didn't come up during negotiations or checks (wonder what the buyer's solicitor was doing when he should have been checking titles).
So because the new owner is occupying the land by permission, he cannot claim possessive title to it.
What possessive title would allow you to do is, for example, move your back fence out and claim a bit of the "waste" land behind it. If the owner doesn't come forward and complain, then after 10 years you can apply for possessive title, but then it is still not fait acompli. There are guides online at (IIRC) the Land Registry that give all the detail.
