How is it regarding a gifted plot of land with house construction?

Whoever is listed in the land register also owns the house that stands on it. Period.

Now, the land belongs to your girlfriend (or will belong to her), if you build a house on it together, then it belongs to both of you - no matter who pays for the house.
If she finances the house alone, that is also fine - if you finance it together, then you have to come up with a solution.

If you are to help finance, meaning you also become a debtor with the bank, the bank will insist that you are registered in the land register with a share - otherwise, you will not get the loan.

Then it also depends on how many shares you are supposed to have in the land register. From the cost perspective, it is cheapest if your girlfriend gives you a share of the land as a gift, under the maximum amount of €20,000 (if you get married, the exemption amount will be higher, but if you are unmarried, it is only €20,000). For that, inquire about the standard land value and calculate how many square meters your girlfriend can gift you that are under the mentioned maximum amount. Then you will be entered in the land register, and there should be nothing standing in the way of financing.
Do this BEFORE you start building, because then only the land price is relevant - if the house is already built, it will be included in the calculation.

However, whether this share also represents the share you contribute to the house is another matter.
If you finance the house equally, then the €20,000 land share gift probably won’t be sufficient to reflect that. Then you can either increase the shares of the land that you are supposed to receive accordingly. Then you would make a gift over €20,000 at the notary and sell the share that goes beyond that. But you have to be careful: first of all, real estate transfer tax will of course apply AND there must be actual proof of money flow - otherwise, it is a gift and will be taxed accordingly (usually higher than the real estate transfer tax).
Another option: stick to the fact that you only own land shares up to a maximum value of €20,000 and regulate a division in case of separation in a contract notarized by a notary. That is also possible and the agreements made there only come into effect if you should separate.

How much is the property alone worth? Of course, your girlfriend can do whatever she wants with the property, gift you 50% or sell you 50%, but what do the parents say about it? What was their intention in giving it only to her and not to you?

Why should parents give a gift to a person who might someday become a son-in-law? It is normal that they leave it solely to the children, and then they can decide how to deal with it.

That is clear to me, however, all parties involved should also be aware that this situation requires more clarification than just transferring a piece of land. Depending on the intention, aren't there even clauses to ensure that the land goes back to the parents in case of separation? But I don't want to discuss that here now.

The point is that a piece of land was gifted, a house is supposed to be built, but one person is left out. Hence this thread. I stick to my point, what was the parents' intention, how did they envision the house construction and by whom?

I have basically dedicated an article on my website to this topic some time ago, which I am simply uploading here. Perhaps a few questions will be answered with it. Otherwise, just ask.

If you are supposed to finance it, that is, also become a borrower with the bank, the bank will insist that you are entered in the land register to some extent - otherwise you won’t get the loan.

That was not an issue for us at all!! Why should it be? Unfortunately, we did not manage to have my husband registered before the start of construction. We only became truly aware later that the matter becomes significantly more expensive after the construction. So if, then carry out the partial gift (x% of the property) to the husband before the start of construction. It can also be problematic if the transfer from the wife to the husband takes place shortly after the gift. In that case, the tax office might reach out. This is then viewed like a gift from the parents and sons-in-law/daughters-in-law only have an exemption amount of 20,000 if I remember correctly.