Flooring for children's rooms: pros and cons

Thank you for your quick and numerous responses!

Carpet? ...

Well, carpet is an absolute no-go for me because you really can’t get it clean anymore. Currently in the rental with parquet flooring without underfloor heating is okay, temperature-wise.

If I were you, I would use a design floor from Meister or Parador for the living area. They do not have plasticizers like vinyl. And in the children's rooms, a high-quality laminate and only then parquet, when the kids are older.
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As far as concreting is concerned – I have actually thought briefly about exposed screed. Regarding Meister, I just read some very bad experiences… And laminate is, from my experience, even more susceptible when it comes to liquids.

It’s just unfortunate if the kids play more in the open living area than in their rooms.

Exactly. How long has your parquet been down and what does it look like now? Is it also more rustic or smooth? (I hope that the damage fits better into the rustic style...)

Would also take carpet, awesome in the kids' room. Also totally low-maintenance and great for allergy sufferers.

Install parquet flooring and lay a carpet in the typical play areas. Our boys, for example, loved the carpet with the streets for a while. A few "flaws" in the parquet do not make the flooring look worse. We were 4 children, the parquet in my parents' house is still great, it has been sanded down once and is now 49 years old. It was not only played on but also celebrated on - which is not necessarily less stressful.

How concerning is vinyl actually regarding plasticizers?

I consider it concerning insofar as measurements can be manipulated: joints sealed, unrealistically high air exchange rates. I would pay attention to the Blue Angel certification, especially in bedrooms. In retail, it usually goes under design flooring and not as vinyl. Only recently have there been adhesive floorings with Blue Angel certification. Vinyls vary greatly in quality and feel depending on the manufacturer. Without underfloor heating, I would also choose wood + carpet.

My parents have parquet flooring in most rooms (the house was occupied in 1990. Since then, the parquet has been there) and we are also 5 children (by now only the youngest still lives in the house). Yes, you can see lots of signs of wear but I don’t think that makes it worse... with many scratches you still know where they came from and then, for example, I sit there on the floor with my daughter and try to stop her from throwing toys on the floor and my mother says to her, "oh, look there... your mom caused that scratch herself with XY... so she should just let you play" and things like that... I like it when you can see that houses are lived in.

Carpet in the children's room is rubbish but not because of dirt - there are carpet steam cleaners - but because you can't build well on carpet.

Personally, I don't like laminate.
Vinyl is good, but I would invest in a thick one because even vinyl gets scratches. And not too light - reddish felt-tip pens and really great: colored craft glue... leave pink stains on light vinyl.