In the original paper of Mehta-Seshadri, it seems like they treat the case of zero parabolic degree (i.e. they prove that zero parabolic degree stable parabolic bundles correspond to irreps of the fundamental group of the Riemann surface minus some points). But, as in the Narasimhan-Seshadri theorem, is the nonzero degree case obtained by considering representations of a central extension of the fundamental group? I'd be grateful if someone may point to a reference.
I think what you want probably follows from Theoreme 2.5 of the paper: Biquard, Olivier: Fibrés paraboliques stables et connexions singulières plates, Bull. Soc. Math. France 119 (1991), no. 2, 231–257.