Install underfloor heating IN wood?

We are switching - like many others - our heating system from oil burner to heat pump. In the newly renovated upstairs, we have underfloor heating and we find it comfortable. However, on the main living floor, there are regular radiators (installed just 4 to 5 years ago). The living room has solid oak parquet, 20 millimeters thick. Also practically new. For the energetically sensible conversion to underfloor heating, I am considering how to do it smartly. Because of the space-saving, there is the elaborate trick of milling grooves for the pipes into the floor. Now I am wondering whether this "milling method" - possibly even easier - would also be feasible in wood. I assume that's nonsense, because wood insulates well and the pipes are supposed to naturally give off their heat to the right and left. Normally, the pipes are cast into a cement surface, and I wonder if grooves in wood are suitable for underfloor heating, where they would presumably also have to be cast in. On top would come the oak parquet again, only much thinner (available, in 8mm thickness, we also have that under the roof). Too much wood with too little mineral, I assume. Two materials that don't really go well together anyway. But I am not done with this thought yet and would appreciate ideas about it.

I assume that's nonsense,

Yep. That's how it is. For so many reasons that you can't list them all without writing a novel. Even if the parquet floor is still so nice, I would first operate the heat pump with radiators. That works well and you only need to adjust the radiators.