Vinyl in floor-level shower

If the moisture problem is targeted here, it is no different than with other materials. However, the dew point does not depend on the material but on the humidity and air temperature, and whether condensation occurs additionally depends on the surface temperature (it must reach the dew point).
So, for example, in summer at 24°C room temperature, the dew point at 60% humidity is 15.75°C. The surface should therefore not fall below 16°C. As far as I know, a normal underfloor heating does not reach below 18°C. At 27°C and 80% RH (after a nice summer thunderstorm), the dew point is already at 23.25°C, which could certainly happen occasionally. Modern systems, as far as I know, have a dew point monitor and automatically reduce the cooling capacity up to shutoff.
It should not go unmentioned that wood is generally more sensitive to moisture (under prolonged exposure to moisture) than tiles or PVC.
But in practice, this is manageable. In general, not too much should be expected from the cooling function of a heat pump heating system, and rather summer heat protection (shading) and additionally air/air heat pumps (colloquially: air conditioners) should be relied on.

It could also be spread through urban legends, implying that wood simply has poorer conductivity. So even if it is cold, it does not feel as cold to us. I would consider this rather an advantage (no cold feet). The cooling effect is not meant to be achieved directly on the skin of the soles but by slightly cooling the air sweeping over the floor. But since air conducts even worse than wood [0.12-0.25 W/(mK) vs 0.025 W/(mK)], this is not the bottleneck and is negligible, especially with bonding (no air layer as an insulator between screed and covering).

At least I was able to convince my wife to have tiles in the bathroom.

By the way, you are to blame for reconsidering vinyl and thinking about parquet. In my parents' house, they had a very sensitive parquet that became very ugly from discoloration and wear.

So what do you recommend? Lacquer sealing or oiled?

I had lacquered floors in my condominium. That stays fresh for longer at first, but then suddenly looks terrible and can’t be touched up so well in spots. When it’s time after a few years, usually everything has to be sanded down once and then lacquered or oiled again.
You might have to maintain oiled floors more often, but that can be done spot-wise if necessary and you don’t have to sand everything completely. You do that only after 15-20 years or more, when there are really too many marks.
But wait for , he’s more the parquet expert here with more experience of different types and with DIY work. Unfortunately, I had to watch the money closely when building the house. That’s why I ended up with tiles downstairs and laminate upstairs.
The parquet in the condo wasn’t particularly good either (rather soft and very thin wear layer), but the lacquered surface still looks good in the spots that aren’t damaged after 9 years.
If I were to redo the upper floor with parquet under the right circumstances and budget, I would oil it. I like it better visually, it’s better for the wood in my opinion, and fits more with the “parquet lives” perspective I described above... And I would choose a naturally rather lively wood pattern. That way, marks (and stains ;) ) aren’t so noticeable... We (=my wife) wanted a very light and calm pattern and that’s biting us now. It just doesn’t fit our lifestyle at all. But you might think differently.

There is also bamboo parquet.

Great! I hadn’t thought of that at all. A colleague at work installed it in his house 25 years ago and raised 2 children there. The floor was sanded once and is in tip-top condition. I don't remember how it was finished, I think it was just oiled. Definitely follow up on this idea.

You have to make sure to use bamboofiber parquet. However, it has the disadvantage that it basically only comes in light or dark. Something like pine white or so does not exist. It has to do with the dense fiber and the absorbency. There is bamboo parquet glued horizontally or vertically, which is more colorable, but by no means as hard.

What do you advise? Paint sealant or oiled?

Oiled / waxed. Always looks natural