The 14-day limit only applies where the parking contractor has had to get your details from the DVLA alone, rather than when you've had a penalty notice affixed to your windscreen (in which case longer time is permitted for the company to then chase the matter up with you).
However if the signs say that if, by entering the car park with your vehicle, you're entering into a contract with Highview Parking (which, in itself would be perfectly reasonable), it's hard to see how APCOA Parking can then raise a charge against you based upon a contract with their company (since there were no notices advising you of any such contract). The two companies appear to be entirely separate, with no connection bewtween them at all.