Can I lay a parquet on top of the tile or do I have to remove the tile?
If so, can it be done on top if the tile is not 100% level. I haven't measured them, but it looks like there are some slight irregularities and the screed is a little sunken in the middle (see photo). We also want to replace the carpeting on the first floor with hardwood flooring or vinyl. The carpet can be easily removed, but only the pile. There is still a layer of glue/foam underneath. Can it be laid over it or does it need to be completely sanded down?
I would knock the tiles out, sand the glue off if necessary, apply a leveling compound, and then lay the parquet.
Even if you're knocking out tile, you need to see what kind of parquet you can use so that its texture isn't too high compared to kitchen tile.
If it's going to be more than a temporary solution, it's better to remove the old floor, prepare the subfloor and lay the new floor.
Carpet as impact soundproofing – sounds appealing at first. However, my feeling is that you should be wary of dampness and mold.
Leave the carpet as soundproofing.
I don't know about tile. If you use vinyl tiles, you will need to do a liquid screed.