This question is currently working it's way through the courts so in 20 years, we might be able to revisit it and offer a very different answer. At this moment, though, the answer goes back to the 10th Amendment: The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.

Education is not mentioned in the Constitution and therefore, is a matter left up to the states. Some states, not all, have state-provided education written explicitly in the state's constitution but that doesn't necessarily make education an constitutionally-protected right. In New York state, for example, the state has a constitutional-obligation to provide an education for every child in the state and Several of legal arguments pushing for education to be a Constitutionally-protected right rely on the idea of it being an unenumerated right (covered by the ideas of the Constitution, if not the exact wording). u/thewimsey explained the concept in the mega-thread on the history of abortion in America:

Simplifying somewhat, an interpretive doctrine called "substantive due process" holds that the right to due process includes not just the right to have fair procedures to determine legal rights (that is "procedural due process"), but also protects certain non-enumerated substantive rights. One of the non-enumerated rights protected by substantive due process is the "right to privacy" - which, despite its name, doesn't mean the right to be free from surveillance, but instead means the right to make certain very personal decisions free from governmental interference. This was first enumerated in Griswold v. CT, which dealt with the right to purchase contraceptives. (It's the right to "privacy" because it's a decision made in private, basically).

Non-enumerated substantive due process rights are famously found in the "penumbras and emanations" of existing rights. A kind of unfortunate phrase because, seeing it in the abstract, it seems to be just made up; in reality, there are a lot of "implied rights" in the constitution, which almost no one objects to. The right to bear arms implies a right to possess ammunition. Freedom of the press implies the right to do certain things in connection with the press, such as sell or distribute newspapers. The constitution regulates the army and navy, but never mentions the air force...but its implied that the US is allowed to have an air force.

In other words, one argument for education as an unenumerated right is that an education is essential not only for understanding one's rights but for liberty.

That though, is only part of the challenge with the matter of education as a constitutional right. The heart of the matter is the nature of that education - and that's the tension that's being argued right now. That is, does a child have the right to learn to read? To learn science? To learn a foreign language? What if their parents disagree with teaching their children science? Or learning a particular mathematic algorithm? What is the state's responsibility regarding the buildings where that education takes place? Etc. etc.