Which parquet can you recommend?

Varnish is common. You don't have to imagine a thick, glossy layer underneath. There are also matte varnishes; in my opinion, a layperson hardly sees a difference compared to oil. Oiled tends to be a bit rougher on the surface.

By parquet, people nowadays mean click parquet with a few millimeters of real wood. It comes ready-made with a surface finish. Little work for professionals, who would certainly like to bill for a few more days of finishing ...

We have solid wood, no click, no two-layer system

There is basically nothing wrong with lacquer, not even on solid wood. The parquet in my parents' house is made of solid wood with lacquer. It withstood four children excellently. When the youngest daughter moved out, it was sanded down and then oiled - which has proven to be less durable.