I am not at all sure about the following since our strata insurer did pay up for what I thought was trivial damage from a small falling branch from an ordinary tree. However, I would have thought that the logic would go like this:
1) Was the thing that was damaged something that was insured against insurable events? I’d say no. The strata buildings and associated structures and landscaping should be covered by the strata insurance for damage caused by insurable events such as impact from things (falling branches, lightening strikes, meteors etc). Somebody’s car is not part of the insured strata property.
2) Would it be covered by the strata insurance public liability component? I’d say maybe. Liability insurance doesn’t pay out for every unfortunate occurrence. First you have to be found liable. If somebody falls down the stairs because they were simply clumsy, that is unfortunate but not the strata’s fault. However, if somebody falls down the stairs because the carpet was worn and loose and created a hazard that the OC should have repaired, then the OC might be found liable for the injury, and the liability insurance would prevent the OC becoming bankrupt.
So, if the tree was in poor condition and in decline and particularly if an arborist had already advised that the tree should be pruned or removed, then the OC might be liable, and the liability insurance component might pay.
If it was just an ordinary tree and there was no reason to suspect that it presented any more risk than any other ordinary tree, then I would suggest that the OC is not liable if someone parks under it. It’s just bad luck that a branch fell on the car. That a branch fell and did some damage is not the OC’s fault so liability insurance would not cover it.