I've recently had a similar experience with Purple Parking, the only other chauffer parking service at Heathrow. On this occasion, they produced photos showing the car when it arrived at their depot, complete with huge scratch down the side. This, they claimed was proof that I had dropped off the car damaged. Of course, there was no way their driver could have damaged it on the way to the depot.
Fortunately, my company does a large amount of business with them, and when we threatened to withdraw our customer they agreed to fix it "as a good will gesture".
I've stopped using them now, and instead pre-book parking at the Pink Elephant carpark through baa.com, which gives you WorldPoints and approx 50% discount on the gate rate. At least then I get to keep my car keys!