A new supermarket has open nearby and it has a large pubic carpark under a part of the building. What is the correct way to map this? Should I overlap the building and carpark and then use the layer tag? If so, what is the correct layer tags? The carpark is at ground level and the supermarket above it.
I suspect you are confusing the supermarket with the building that contains it. Near me, it is quite likely that the building will also have flats (apartments) above the supermarket, all in one building.
You should probably be using level, rather than layer, with a separate object for the building outline. https://wiki.openstreetmap.org/wiki/Key:level
Take some inspiration from https://www.openstreetmap.org/#map=18/50.17750/8.62863 (Riedbergzentrum). An entrance to the underground-parking is at https://www.openstreetmap.org/node/2388950086 .