I think there are some pretty big misconceptions here about the different products used.
There is a world of difference between applying a thin layer of a pre-manufactured adhesive from a bucket - as in the case of the the thin layers of adhesive used to fix the Marley and other (often asbestos containing) tiles so commonly used from the 1950s to the 19
80s (note that latter date, adhesives containing asbestos were in use until the 1980s and usage was legal until the end of 1999) - and the thick layer of semi-molten bitumen that parquet was traditionally bedded into, where the parquet blocks are pushed into placed then hammered down to ensure a good bond. Because that process requires the heating of lumps of bitumen in a boiler to a molten state, incorporating asbestos - a heat insulator - would be counter productive, slowing down the process and increasing the cost of the gas used (if you've ever seen a bitumen boiler on the back of a lorry when they are doing road repairs you should be able to gauge the amount if heat generated by the boiler, and therefore the volume if gas required to do the job). In service a thick layer of adhesive is less likely to need an form of fibrous binder. That is one reason why you are highly unlikely to find asbestos fibres in a parquet flooring bitumen layer
Tile adhesive was often laid cold and incorporated fibres to help improve the cohesiveness of the product (in the same way that nylon or UHMW fibres are used in some concrete mixes) - in other words the fibres allow the installer to spread a very thin layer of adhesive which will resist breaking up even if the substrate, such as concrete, is moving as it cures. Parquet, on the other hand, a much older process, depends on having a thick layer of "adhesive" onto which the block are bonded
Some sections of the parquet have subsided. The contractor that I spoke to had reservations about lifting those sections to level the sub floor due to the bitumen containing asbestos.
Then find yourself another contractor, one who has experience of dealing with parquet. It isn't a common skill any more, but it hasn't died out completely. BTW the blocks don't "subside" - they can get worn over time
I'm not sure at this time how we could make it level again.
How bad is it, though? Are we talking about 3mm hollows, or 15mm hollows, here? Have you checked this with a long straight edge, an inspection lamp and some shim packers? 3mm hollows will sand out with a floor sander in many cases. For bigger differences a competent parquet flooring man will lift out any really low sections and also any damaged blocks and replace them (along with reinstating missing blocks) using matching recycled or brand new blocks, cut to size. It isn't rocket science and a comoetent DIYer should be able to do the job as well. The modern adhesives used are cold set glues and do not require a bitumen boiler any longer. Once the replacements are in he will then sand the floor flat.
Do a search of this forum for references to LeCol, one of the main adhesives used in the refurbishment process and you'll find greater detail. This isn't a cheap process, but the end results can be stunning when the job is done right. There are less and less of these floors out there, at least partly because some contractors can't be a***d to deal with them
