This is a common misunderstanding.

Rights are unconditional is the sense that they apply to everyone equally. But almost all are conditional in the sense that they don't give carte blanc permission without exception.

It might help to think of rights like axioms. (A & (A -> B)) -> B is true unconditionally. It is an axiom. But it is also conditional. It is universally true, and yet, B is not universally true.

Nearly all rights we have are unconditionally conditional in this way. They always apply to everyone (universally), but that doesn't mean there aren't conditions (that get applied universally).

For instance, being convicted of murder triggers conditions on most of the rights we have as citizens. Not all of them, but most.

Privileges are different from rights because they don't have to apply to each person equally in the way rights do. The two are separated by the fact that rights are extended universally and privileges aren't, not by the presence or absence of conditions.