Remove tiled floor or lay parquet/vinyl on top?

I would do it the way the tiler says, tile and parquet on top. Everything else is unnecessary waste of money and time. And it creates a lot of pressure, produces construction debris with disposal. Regarding underfloor heating, tile on tile would of course be more efficient. But it should work this way too. The heat then just has to pass through an additional layer, and with parquet (wood, which insulates) some heat is lost. You need to be aware of that. Otherwise, from my point of view, there is no reason not to simply go over it. The tiler will have checked the old tiles. If they are firmly fixed, they are a good base for another layer. The only consideration on my part: If the underfloor heating should be replaced anyway (age? efficiency), then it would be worth considering doing a clean sweep and rebuilding everything from scratch.