Unless something is explicitly forbidden in law it is allowed, so why
do unitary states have constitutional rights? There is no level below
them that could mean differently. Nothing would counteract a
constitutional right if it was removed.
Any state with a constitution of any form has rights that arise from it. Any kind of law establishes rights and responsibilities, and constitutions are not an exception to that rule.
In Europe and other places with a constitutional tradition of parliamentary sovereignty largely unfettered by judicial rule, the primary mechanism by which human rights are protected that is resistant to ephemeral or bare majority parliamentary action, and which is enforced by courts, is through international human rights treaties. In Europe, these were devised and administered largely by an international organization parallel to the European Union called the Council of Europe which promulgated and secured widespread adoption of with little qualification, the European Human Rights Convention.
Most Western democracies, unlike the United States, provide that a conflict between a duly adopted treaty that remains in force, and subsequently adopted domestic legislation, is resolved by the courts in favor of the treaty. These treaties are the strongest protections of human rights in most of these countries that are enforced in their own domestic courts and regional international human rights courts which countries in such treaties take more seriously than the U.S. does.
There is circumstantial evidence from the structure of the U.S. Constitution that the drafters of the 1789 Constitution of the United States intended the same thing, but the Courts promptly settled on the opposite rule flowing from a very literalist reading of Article III of the U.S. Constitution that doesn't consider intent or context and now has the force of precedent behind it.
While it is much easier to amend constitutions in most unitary states (which are frequently amended much like U.S. state constitutions), many still do have human rights protections in their constitutions, largely as a result of a movement when constitutions were redrafted following World War II which interrupted most of Europe's national regimes at least temporarily due to Nazi occupation or rule, on the model of the United States Constitution and as a way to honor their new membership in the United Nations by incorporating much of the U.N. Declaration of Human Rights (a non-self-executing instrument) into enforceable domestic law, usually supervised by a separate constitutional court. As recent shifts away from a consensus around those rights have eroded, treaty rights have proven more important than constitutional rights (e.g. in Poland and Hungary). But inertia, and the popularity of the general idea of protecting constitutional rights even though particular aspects of those rights may be unpopular, has kept those protections on the books as a political matter even though they wouldn't be hard to repeal from a process standpoint.