I don't really understand your point. The person who has to deal with this is the Keeper. Anyone can deal with it for the Keeper but it has to be done in the Keepers name.
The driver is the only person actually liable but they have no idea who the driver is unless the Keeper blabs the identity of the driver, inadvertently or otherwise. If their Notice to Keeper is 100% fully compliant with the requirements of PoFA 2012, then they can transfer liability from the unknown driver to the known Keeper.
So, without understanding who person A or B is in relation to the Keeper/Driver, it means nothing. Just in case there are two different drivers involved, the liable driver is the one who drove in and parked. But they still have no idea who that is unless the Keeper tells them.