The card voucher(*) for the transaction will have the last four digits of the card number, the type of card credit/debit visa/mastercard/why and if not contactless the expiry date and maybe the start date. Fairly minimal chance of a random voucher matching all those with the card in your wallet. Even without the voucher or goods receipt you just go to your card provider and ask for the full information on that transaction.
They don't have a foot to stand on, let them take you to court and end up paying their and your costs.
The letter from the parking co will have been sent to the registered keeper of the vehicle they think has committed an offence, sourced from the DVLA via the vehicles registration number.
(*) The "voucher" information may or may not be on the main receipt for the goods, it varies across stores, aldi and co-op have a seperate vouchers, tesco is all together, sainsbury's are double sided so effectively all together.