Can we add a floor to our parents' single-family house?

You have written nothing about it - among other things, that's why I pointed out the missing information that there is already a staircase leading to the attic. So I assume that it still needs to be constructed.

Building expert on the go

I would first inquire at the building authority whether you can get something like this approved ([Bebauungsplan], [Nachbarschaftsbebauung], etc.) It doesn't cost you anything, but you will immediately know if it's worth considering.

I would first ask the building authority if you can get something like that approved (development plan, neighboring buildings, etc.)
That doesn’t cost you anything, but you’ll immediately know whether it’s worth thinking about.

The problem is that they always want a finished plan from the architect, otherwise they won’t make a statement. That’s why I want to inform myself as well as possible beforehand whether that is even an option before I then plunge into costs "for nothing"...

But they must be able to give you a definite statement about what is allowed and what is not. Normally, at least, it’s not just randomly decided...
If I were you, I would first go to the building authority and sit down with them. Then you can quite quickly see what is possible without any special permits. With this information, you can then go to an architect.

If the architect then believes that the existing requirements are compatible with your wishes, you can still decide whether you want to do it all or not.
Until then, no excessively high costs should have been incurred.
In such cases, you usually have to invest a small amount first to find out your options.

Regarding the neighborhood, there are identical houses in the immediate vicinity

I would proceed as follows:
Look into the development plan or ask the building authority how high building is allowed and whether the floor area ratio plays a role. Then there would also be the question of whether a two-family house is permitted and whether the parking spaces need to be increased. Access and feasibility etc. can first be assessed by oneself. Then bring an engineer and construction experts on board. That costs money, but not huge sums.
Such a professional should also be able to roughly estimate the costs.
In terms of price, I see the costs like a bungalow without a floor slab. Added to or deducted from that would be the recycling of the roof and the access to the living level.