Ethel basically answered your question in the first post above. To be legally pedantic, Asda can't actually 'fine' anyone but they can pursue them for a civil debt.
The signs stating that (with certain exceptions) parking will be charged at �60 per visit constitute an invitation for drivers to enter into a legally binding contract. If a driver then parks in one of the spaces, his action in doing so signifies his agreement to the contract. (i.e. the act of parking has the same effect as signing a written contract).
From that point on, the driver is committed to paying �60 in settlement of the contract. If he fails to do so, the supermarket can pursue him through the civil courts for payment of the �60, together with court costs. (Obviously, they must write to the driver first, to offer him the opportunity to pay before court action is taken). Although DVLA will not release details of names and addresses of the registered keepers of vehicle to casual enquirers, they are permitted to provide the information under such circumstances.
However, it would be interesting to see what would happen if a driver, who had received a �60 bill, responded by saying 'Yes it was my car but I wasn't driving it'. There is no provision within civil law to force the registered keeper to reveal the driver's identity. Neither is there any way to force the registered keeper to pay a bill resulting from a contract which another driver has entered into.
Chris