Regarding the natural=wood tagging, while it is not in line with the initial meaning of the tag, in practise it is now very common, even so that the wiki definition has been adjusted. The tag landuse=forest is also widely used for the same purpose (tree covered areas) and equally questionable, from a semantic point of view, landcover=trees would be better, but as the devs of the osm-carto rendering style (aka “main map”) have decided to ignore the landcover tag, it will not result in satisfactory rendering and people are not adopting the tag (or are double tagging, which leads the carto team to telling that the tag is not needed because the things render also without it).
To summarize, I would keep these tree covered areas as this use of the tag is common practise (from a data perspective, one can see that they are too small to be actually “woods”, so there is not much of harm about it).
For the parking lots, amenity=parking with the parking=* (multistorey, surface, underground, etc.) specification, should be applied to the whole parking. For individual parking lots, amenity=parking_space is there. Eventually it could be used together with capacity to map rows of lots
