You don't mention the age of the 'new' house....
It depends on the floors construction. Standard is to have a 50 to 75mm screed on top of a 100mm ish concrete subfloor on top of 150mm crushed and compacted hardcore/stone. Insulation has only been added since the 1990's.
If so, you'd need to remove the whole screed and hope that it's not terribly bonded to the concrete subfloor (sometimes is, sometimes not).
But I'm worried about any house old enough to have original parquet flooring. It's possible that the little wooden blocks are glued straight on on top of a concrete subfloor. Then you're in trouble.
Removing or digging out as you say to a specific depth in concrete is difficult. It's usually easier to remove the thing and start over.
Also heating a floor without insulation underneath is pretty ineffective, again it's best to... Remove the floor, dig out, insulate, new membrane, new concrete, heating pipes set into a new screen. Floor covering of your choice.
A lot of work. However there are mechanical scraper devices for removing sticky substances/adhesives, but I'll not lie to you, whichever way you do it will be hard work and take a while. Depending on age I'd have some of the black adhesive tested for asbestos too, as it's possible.... (but don't panic, just don't create dust whilst working on the adhesive. Tests are cheap, £30 or so)
Re: new info/comment...
Yes, insulation stops the heat going the wrong way if you like. There's a couple of schools of thought, one is to insulate directly under the heat source, throwing all the heat upwards and some like to insulate under the 'mass' if you like, then the heat soaks into the mass (concrete slab say) which acts much like a storage heater, evening out heat fluctuations etc.
You'll struggle to dig out the middle because of sealing the membrane etc.
Thinnest/cheapest way IMHO is to (either follow what AndyT said) or glue back any loose blocks roughly, sand the whole thing flat (again roughly) and then fit a new laminate floor over a heating mat. Try to get both from the same company for best reassurance. Pergo 'quickheat' is as good as any.
You'd probably get it all in around 22mm or so.