How to tag parking decks at ground level with 1 or more underground levels

So far I’ve effectively ignored the underground part, just mapping the surface as were it a parking lot plus the entrance ways to the first underground level tagging those with level=-1 like this, KISS so to speak.

access=yes
amenity=parking_entrance
covered=yes
level=-1
lit=yes
parking=underground
smoothness=good
supervised=no
surface=concrete

Reality is the surface level parking is actually the roof of the underground parking thusly in effect having a 2 level parking. Started to tag a multi-storey but then all sorts of issues started cropping up like all the ‘roof’ ways are technically location=roof, parking spaces including, building:levels=0 + building:levels:underground=1.

Question is, to tag as building=parking + parking=multi-storey (or would it be multi_storey?), or let the building notion be and get into parking:levels=2 (found a few dozen entries on TagInfo), map the surface details and let the underground entrances be just that?

There’s actually quite a few around, surface parking with level(s) below, think a German discussion sort of having a consensus to not tag this as a building, and that is OK long as there’s none up top when many just have a garden/park type idea on the ‘roof’ at street level.

A headache dossier (for me).