If it's true that the Asda car park will be closed to the public then it may be that they need easy access for their teams of shopfitters etc. Shoppers will just have to park in one of the council car parks. They only ever seem to be full at Christmas. Or people could always walk.
I would expect Asda to introduce a charge for their car park anyway. Refundable after spending something like £5 in store. Afterall Asda will probably be considerably busier then the Co-op, so they'll need their car park for their customers, not for other local traders.