ASDA did use Smart Parking on most of their stores (kicked out April 2016) but if they are on a shared site, it could any parking contractor. There is little point in contacting Smart about an ASDA ticket now.
Best now to check with the store about appeals except where you see a sign and then use the email address instead. All ASDA stores have a "genuine customer" waiver so send copies of receipts for shopping there. If you don't have receipts for the day send bank statement showing regular shopping there. It usually does the trick.