We would suggest starting with a close read of the terms and conditions of the meet and greet service. It is possible that one of the conditions is that when collecting the vehicle, signing for the keys or leaving the car park is deemed acceptance of the vehicle's condition and that any damage must be reported at the time. If this is the case it will be difficult to get them to accept that they are responsible for the damage, as you will likely need to provide evidence that shows the vehicle was already damaged when it was handed back to you. We would suggest following their complaints procedure, and if this is unsuccessful to escalate to a parking trade body such as the British Parking Association or the Civil Aviation Authority.