Your takeaway point is always good advice - always book the full extent of the 24 hour period you've paid for.
The parking system is all automated, so I don't imagine they'd have any process for manual intervention. From their perspective, it is a parking space that should be earning them money being filled...
Normally, I would have thought this would be part of the costs to recover from the airline or possibly travel insurance, if there's no luck with Heathrow themselves. That said, I'd expect a goodwill gesture on their part to waive the cost - ultimately, Heathrow failed to keep their airfield open. The lack of recognition of weather in airline compensation is a source of confusion and frustration for me!
I assume their assertion of "recover it from the airline" would be because Heathrow would have already paid the compensation to BA for the delays incurred?