Brauer group of moduli stack of stable parabolic $\textnormal{PGL}(r)$-bundles over a curve

Let $k$ be an algebraically closed field of characteristic zero. We prove that the Brauer group of moduli stack of stable parabolic $\textnormal{PGL}(r,k)$-bundles with full flag quasi-parabolic structures at an arbitrary parabolic divisor on a curve $X$ coincides with the Brauer group of the smooth locus of the corresponding coarse moduli space of parabolic $\textnormal{PGL}(r,k)$-bundles. We also compute the Brauer group of the smooth locus of this coarse moduli for more general quasi-parabolic types and weights satisfying certain conditions.