Can I use my parent's house as accommodation for a Schengen visitor visa?

After calling the local authority, I managed to clarify how to prove my parent’s house as accommodation. I had to fill out a document called "befogadó nyilatkozat" (which means something like "hosting declaration"), have my parents, who are the shared owners of the house, sign it and have 2 witnesses sign it as well. In practice, my brother and I were the witnesses. With this paper, I was able to obtain the official letter of invitation, with which in turn, my girlfriend was able to obtain her Schengen visitor visa.

I hope this will help someone but do note that the process seems to vary between different Schengen countries, my example is specific to Hungary.

Edit: as of 2023, this two-step process for Hungarian visit visas and temporary residence permits has been simplified, so now you don’t need to get an official letter of invitation. Instead, you have to send all the supporting documents, including the signed "befogadó nyilatkozat", to the applicant, and they have to submit these in their application.

just using the letter of invitation as proof is not applicable for me

Well yes it is. Proof is transitive. For example:

• You have to prove your identity to get a passport, so a passport is proof of identity.
• You have to prove that you know how to drive to get a driver’s license, so a driver’s license is proof that you know how to drive (in theory, at least!)
• You have to prove that you can accommodate your girlfriend to get the letter of invitation, so the letter of invitation is proof of accommodation.

But let’s look at this in a little more detail.

First, do not pay undue attention to anything you read on schengenvisainfo.com, except perhaps the disclaimer (emphasis added):

Legal Disclaimer

SchengenVisaInfo.com is not affiliated with the European Union, the EU Council, or any other government agency. SchengenVisaInfo.com is a private publishing website that offers the Schengen Visa Information Guide Online.

SchengenVisaInfo.com does not issue Schengen Visas.

Our guides, news articles, and blogs are for general advice only. We have made all reasonable efforts to ensure that the contents are correct. However, we do not guarantee the completeness or accuracy of the information and shall not be responsible for any errors, omissions, or inaccuracies. Any or all of the content is only for informative purposes & is in no way legally binding.

The travel restrictions in Europe change without notice and all travelers should contact their own country’s embassy or the Schengen consulates in their home country before planning or booking any travel to any EU country.

Our general advice should never replace that of any government organization or embassy/consulate.

Furthermore, that site has a very poor reputation among users of this site because it has frequently been out of date by several years.

The information you’ve found there describes general rules about what each Schengen country may require, not specific information about what you must provide. Note the weasel words (emphasis added):

Proof of Accommodation is a rule of the visa application procedure that each Schengen country ought to apply to their foreign visa seekers.

Similarly, it quotes the Schengen Visa Code (emphasis added):

Member States may require applicants to present a proof of sponsorship and/or private accommodation by completing a form drawn up by each Member State.

That does not tell you anything whatsoever about what Hungary actually requires (and, sure enough, that text is outdated, having been amended as of July 2019).

Instead, refer to the website of the Hungarian consulate responsible for your girlfriend’s place of residence. This will include a link to https://konzuliszolgalat.kormany.hu/how-to-apply-for-visa, which in turn has this link to a PDF document: Short Term Stay (not exceeding the 90 days period).

From that document, it is clear that, for visits to friends and acquaintances, the Hungarian visa application does not require proof of accommodation in addition to the invitation letter:

When applying for a short stay uniform visa (Schengen C-type), the purpose of the entry can be
certified with the following documents:

…

• the case of visiting friends, acquaintances: a letter of invitation issued and endorsed by the
  National Directorate-General for Aliens Policing or the letter of invitation in the form of a public
  notarial document and a photocopy of the personal identification document/passport of the inviting
  party in Hungary.

The only mention of "accommodation" is in cases that are not relevant to your girlfriend’s visit. Hungary knows that proof of accommodation is required to obtain a Hungarian invitation letter.

With this in mind, let us turn our attention to your question:

how can I prove that this house can be her accommodation?

Follow the instructions that you linked to above for applying for a letter of invitation.