If it's not addressed to you, you can't appeal it. The person who received the PCN needs to appeal or name you as the driver and transfer liability to you.
The signs are fairly clear the car park is for customer use therefore as you can't be a customer if the store isn't open you'd be breaching the terms. Not to mention the point about needing to register. Many car parks have no way to 'pay' and you get 5 minutes to get in, read the signs and leave, that's the consideration period which given Aldi car parks aren't big and was likely empty at that time of night should have been enough time.
You haven't shown us the back of the PCN although I suspect it is POFA compliant, it was given in the correct time frame and I've noticed they've changed the wording g to invite the keeper to pay.
Either way at this point not your problem unless the keeper tells them you were driving, then the for appeals resets. In that case you will need a stronger defence than I tried to payo but couldnt. Looks like a building site on GSV so I suspect a new store with lots of shiny new signage. You can even see the signs warning new rules in place in your photos.