Simplest way is to claim from your own insurer and let them decide who to try to try to recover from if they can identify which vehicle caught fire first, and can find some evidence of negligence on the part of the owner. I think the LR that started the Echo Arena fire had a fuel leak that the owner was aware of, but this one may not be so straightforward!
I spent over 30 years in the insurance claims industry and can't see a claim against the car park operators getting anywhere. Car Parks aren't required to have sprinklers, that's just a recommendation for the future but sprinkler installations don't come cheap and only operate effectively if there is an adequate water supply.
My last employer was big into loss prevention and ran their own test lab/research facility in the US. When they took a lease on a new UK office in 1999 we didn't move in until sprinklers had been installed, but a few years later a pressure test revealed that there wasn't enough mains pressure for the system to operate effectively, so a massive water tank got built in the car park that was the same height as the office block!
It will definitely be messy that's for sure and will drag on for years like the Echo Arena one did.
The Buncefield fuel depot fire happened in 2005 and it took insurers, including my employer, until 2010 to get a judgement!
Although when I left in July 2010 they still hadn't received any money.