Let PMα s be a moduli space of stable parabolic vector bundles of rank n ≥ 2 and fixed determinant of degree d over a compact connected Riemann surface X of genus g(X) ≥ 2. If g(X) = 2, then we assume that n > 2. Let m denote the greatest common divisor of d, n and the dimensions of all the successive quotients of the quasi–parabolic filtrations. We prove that the Brauer group Br(PMα s ) is isomorphic to the cyclic group Z/mZ. We also show that Br(PMα s ) is generated by the Brauer class of the Brauer–Severi variety over PMα s obtained by restricting the universal projective bundle over X × PMα s .