Let S (respectively, S′) be a finite subset of a compact connected Riemann surface
X (respectively, X′) of genus at least two. Let [Mscr ] (respectively, [Mscr ]′) denote a moduli
space of parabolic stable bundles of rank 2 over X (respectively, X′) with fixed
determinant of degree 1, having a nontrivial quasi-parabolic structure over each
point of S (respectively, S′) and of parabolic degree less than 2. It is proved that [Mscr ]
is isomorphic to [Mscr ]′ if and only if there is an isomorphism of X with
X′ taking S to S′.