  We find all possible isomorphisms and 3-birational maps (i.e., birational
maps which induce an isomorphism between open subsets whose respective
complements have codimension at least 3) between moduli spaces of parabolic
vector bundles with fixed degree. We prove that every 3-birational map can be
described as a composition of tensorization by a fixed line bundle, Hecke
transformations, dualization, taking pullback by an isomorphism between the
curves and the action of the group of automorphisms of the Jacobian variety of
the curve which fix the r-torsion. In particular, we prove a Torelli type
theorem, stating that the 3-birational class of the moduli space determines the
isomorphism class of the curve.
