Law.SE encourages answers to a question asked with respect to one place's law under the laws of other jurisdictions with which the person providing the answer is more familiar. I am doing that here, despite not providing any answer concerning Greek law.
Another closely related question, which I also don't answer under Greek law, is whether ratification after the fact of your own signature on a document, when not specifically authorized by you in advance and without indicating that it wasn't made by you, is a crime or has other non-criminal consequences. This depends to a great extent on the context in which it comes up. Usually, under the laws of most Western style legal systems, if someone is trying to prove that you signed a document and you ratify it after the fact, this make the contract valid and binding upon you, even if you admit that you didn't actually sign it, and that you didn't authorize someone to sign that specific document in advance for you.
united-states
In the United States, the proper and best practices approach is for someone signing on your behalf with your permission in advance to do so, to print your name and then say "by [their name] as agent of [your name]" followed by their signature of their own name.
The fine details, when this is not done, however, differ in fine detail from one U.S. state to another and based upon the kind of document involved and the context in which a challenged to the authenticity of the signature is raised.
Generally speaking, if this is not done, then, when push comes to shove, it depends a lot on the nature of the document signed, the exact language of any forgery statute sought to be enforced, and the context in which the challenge to the validity of the signature comes up.
In the case of a health insurance contract, the most likely situation would be that the only legal consequence would be that you are bound by the forged signature that you authorized. Even in the context of a health insurance contract, however, exactly what was being signed would matter. A mere signature agreeing to the contract would be less problematic. A signature representing that facts are true under penalty of perjury or in a notarized affidavit or acknowledgement, would be more problematic and might be a minor crime (although perjury requires that a misstatement be material, which one could argue that it was not, if the signature was authorized by you and you would have been able to sign and represent that all facts you swore were true were really true).
In the case of real estate documents in New York State and some other jurisdictions, there is a principle called the "equal dignities law" that requires any document authorizing an agent to sign on your behalf to have the same level of formality (i.e. a signed writing and notarization) as the underlying document being signed by the agent on your behalf. This minority rule doctrine is designed to prevent verbal authorizations for someone to sign on your behalf and to prevent unnotarized powers of attorney from being used to sign a document that requires a notarization.
For some government or official documents, a signature without an acknowledgement that it is signed by someone else could be more problematic. There would often be statutes criminalizing this in the case of very specific kinds of documents on a case by case basis.
It is almost always a crime in the United States for someone not identified in the notarization by name, either as an agent or actual signatory, to sign the notarized document.
A comment by WeatherVane to another answer notes that the abbreviation "pp" by a signature, meaning per pro was routinely used in U.K. professional correspondence by a secretary who had been instructed to send a letter, in the absence of the writer. When this was done, the Secretary made no attempt to copy the writer's signature and no POA was needed. This practice was never widely adopted, however, in the United States, and draws from a common law legal tradition that is not shared by Greece.
europe
I'm not familiar enough with the laws of Greece to be comfortable that the way that this issue is analyzed in Greece would be the same as it is under United States law.
Greece has a civil law system in which the whole body of law regarding contractual formalities like the necessity of signed writings, the disclosure of agency relationships, the notarization of documents, and the formalities involved in executing government documents, which differs materially from the law on those subject in common law countries like the United States and England.
For example, until 200-300 years ago, one of the formalities involved in a transfer of real estate ownership in the common law world was the "livery of seisen" which involved the delivery of a handful of dirt from the seller to the buyer, something that was never part of the law in civil law countries and derived from traditional Celtic and/or Germanic tribal customs. Civil law legal traditions, in contrast, derive largely from Roman law which fell into disuse when the Roman Empire fell in the 5th century CE, but was revived and "received" as a source of legal authority again on a widespread basis in the early modern period starting in about the 15th century CE. This process of reception was then standardized and modified for local and modern practice with the adoption of civil codes starting with the French Civil Code, in the early 19th century CE, and the Greek Civil Code was part of this movement spurred upon by Greece gaining its independence in the Greek Revolution that began in 1821 and was adopted as part of the process of establishing Greece's national identity at that time.
Greece's law on these topics would be much more similar to the law of other civil law countries in Europe and Latin America. But on this specific practical issue concerning contract execution formalities, I would not even been strongly confident that Greece's law on the subject would be the same as the law in, for example, France or Argentina or Germany. Greek people have been signing official written documents for much longer than almost any other country in the world that still borrows anything from its local legal traditions (with the possible exception of China). And, it is not implausible that some Greek specific traditions have evolved that are not shared by other legal systems whose history of using signed legal document is 2500-2800 years shorter.
I don't have enough confident knowledge of how it is done in Greece in real life to provide an answer to the title question under the designated Greek law.
Historically (going all of the way back to the earliest written documents of the Sumerians, Egyptians, Akkadians, and Hittites), aristocrats, government officials, entity officers, clergy, and certain other important business people used personal or entity official seals to approve official documents in the Middle East, North Africa, Europe and Latin America.
The use of personal seals of individuals was already considered old fashioned, and was an obscure footnote in legal history, a hundred and fifty years ago. The use of signature seals had reached a similar status for entities and clergy, and for pretty much anyone other than notaries and certain government officials, by sometime around fifty years ago (give or take, and varying somewhat from country to country). This historical footnote isn't relevant to the current law in modern Greece, but it is relevant to the ongoing use of signature seals in Asia discussed below.
asia
In East Asia and Southeast Asia (basically, places which were historically influenced culturally, politically, and legally by China), it is common to use official personal seals to sign important documents, particularly in the case of middle class and higher social class individuals who can afford to have a seal made for them, in lieu of a manually signed signature of the person signing a contract or document. It is also ubiquitous for entities in these places to sign important written documents with an official seal of the entity (sometimes informally called a "chop").
In part, this is a product of a different writing system in countries that used Chinese characters, both directly, and because the percentage of people who were literate enough to write their own names legibly with a calligraphy pen and ink was much smaller than it was in Europe at comparable times.
In these countries, possession of someone's personal seal is typically sufficient to establish apparent authority to make a binding signature on behalf of the person or entity to whom the seal belongs, without a separate written power of attorney document.
So, if this issue arose, for example, in Taiwan, rather than in Greece, if your parents had possession of your personal seal, their use of it to sign a contract with your permission would probably not be a problem. But, if your parents crafted a counterfeit seal to sign the contract on your behalf, even with your permission to do so, this would probably be illegal and would probably be a crime.