Parking Eye's evidence to POPLA will include their contract details even if you haven't raised it.
I think you are referring to the contract implied in parking on the land, i.e. that between the driver and the car park operator. I have appealed many cases requesting proof that the car park operator (Parking Eye) has the explicit written consent of the landowner to issues such charges. To date, this evidence has never been presented to POPLA and all the appeals I have submitted have been upheld on this basis.
How did your missus have the car for lunch and ferrying the kids if you were in work all day?
Ignore early advice. Tell Asda. They may at least be able to confirm the two transactions - are the cards on the same account?