Land planning for a triangular plot

Budget missing.
And regarding §34, information about the neighboring buildings (Bauhaus/gable roof) is missing.
You don’t have many options. Just try placing 10 x 10 on the plot. That won’t work.
I would go with an L-shape on the ground floor and then build part of it as a second story over the attic/upper floor. Then it would be a southwest-facing garden.

Regarding the boundary garage: a boundary garage is less of a problem than a two-story building you want to build west of the parents’ garden.

Oh, I thought the "Surroundings" picture would be sufficient, as 11ant already said.

Budget up to €400k total costs, we would get the plot as a gift.

Regarding the surroundings, the parents’ house is single-story with an expanded attic and an unconverted loft space (building height is 8.3 m). The new building east on the other side of the street has two full stories and an expanded attic. On the left edge of the picture, a raised bungalow can be seen. Opposite that, a group of three terraced houses. The house north of the parents’ house is also single-story with an expanded attic, large dormers on both sides, and an expanded loft space.

Regarding building to the west in general: the main view axis from my father’s living room faces south. To the west are the kitchen and the dining room. The old gentleman lives there alone and mainly stays in his study, which faces east towards the street... So actually no problem.
Ideally, I would have liked for him to move out of the house, which is much too big for him anyway (170 m² living space), and for me to move in with my family. Then I would only have to pay off my brother... But you can’t transplant an old tree (his saying).

I don't see two full floors here, otherwise you look here: - also east of Cologne ;-)

I have seen it, but Aachen is west of Cologne ;) however, we need at least one more children's room.

The word "transplanting" for moving something twenty meters seems too dramatic to me, and the good old tradition of the retirement house for the old farmer fits perfectly. The Bauhaus at three o’clock as well as Father’s overnight neighbors at half past eleven with the large dormers I consider legally single-story, even if common sense rubs its eyes there.

Aachen is west of Cologne, though

Caught, the other one on the right. But when I still had hair, I was originally blond.

however, we need at least one more children's room.

That works out just fine with one less oak.

The word "transplanting" for moving something twenty meters seems too dramatic to me, and the good old tradition of the retirement house for the old farmer is exactly fitting.

Unfortunately, this option is not feasible for us cost-wise, as the existing property (built in 1984) would have to be extensively renovated anyway. Building an additional house on top of that unfortunately exceeds our budget. And then there is also a brother who, of course, should not be left empty-handed.

Unfortunately, this option is not feasible for us cost-wise, since the existing property (built in '84) would need extensive renovation anyway.

How does that fit with what you said earlier:

Ideally, I would have preferred if he moved out of the house, which is far too big for him anyway (170 m² living space), and my family and I would move in.

???

Then building an additional house unfortunately exceeds the budget. And then there is also a brother who, of course, should not come away empty-handed.

The retirement house would ultimately also be significantly smaller, otherwise it wouldn't make much sense for the father. Paying out the brother, I wouldn't see as such a liquidity burden: the brother could receive correspondingly more shares of the "Father and Heirs Real Estate oHG" (or something like that) as a gift; as the necessary building land can anyway be leased from you as an inherited lease. Non-nobles constantly commit self-damaging acts of asset disposition "by the dozen," I have the impression ;-(

How does that fit with what you said earlier:

He could, in his old age, simply move into a living unit that suits him better, keyword accessibility.

The retirement cottage would, after all, be much smaller, otherwise it makes little sense for the father. I wouldn’t see buying out the brother as financially straining: for that, the brother could be given correspondingly more shares of the "Vater und Erben Grundbesitz oHG" (or something like that) as a gift; as is anyway possible that the necessary building land can be leasehold inherited from you. The non-nobles constantly commit self-damaging actions to their assets "a dime a dozen," that’s my impression ;-(

Who is the appropriate contact to untangle such a constellation? Preferably via PM, if possible.