Hi there,
Tree owners are only liable to pay for damages caused by their trees if there is compelling evidence to suggest that the tree in question posed a predictable danger. For example, if a tree is very old, dead or partly dead, damaged or diseased, the tree's owner can be held liable for damages caused by that tree. An expert (usually contracted by your insurance provider) should examine the tree to determine whether the tree's owner is at fault.
The owner of the parking lot is also responsible to ensure that the service they provide does not pose a risk for customers. If it is found that the tree in question posed a threat which was ignored by the owner of a parking lot or road, then they may share liability with the tree's owner.
If the tree's owner and/or the parking lot owner are at fault, they or their personal liability insurance provider should compensate you for damages.
If it is determined that they exercised due dilligance in caring for their tree and there was no reason to believe that the tree posed a threat before the incident occured, then you will be left to pay for the damages yourself. If you have semi-comprehensive or fully-comprehensive car insurance, you can claim the cost of repairs back from your insurer - but you may lose claims-free discounts.
Best regards from Moneyguru