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.

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Vamsi, it is hard to understand your question... Maybe you are asking the following: ? Let $V$ be a stable bundle on a curve $C$, and suppose that the degree of $V$ is non-zero. Is it true that one can naturally associate a representation of a central extension of $\pi_1(C)$ to $V$? – Dmitri May 10 '11 at 22:15
If I remember correctly you should find something about that in the volume n.96 of asterisque, "Fibres Vectoriels sur les courbes Algebriques", by Seshadri. – Mattia Talpo May 10 '11 at 22:48
@Dmitri, I mean : Let $V$ be a stable bundle on a smooth curve $C$ and suppose the parabolic degree (=deg(E) + sum of weights of the flags) is not zero. Then, is it true that one can naturally associate a representation of the central extension of $\pi_1$ of $C$ minus some points to $V$? – Vamsi May 10 '11 at 23:12
@Mattia: Thanks. Do you happen to have an online copy of the paper? I can't seem to find it at all? – Vamsi May 10 '11 at 23:14
It turns out I didn't remember correctly, sorry: I just checked and only the degree 0 case is treated. Anyway I don't think an online copy exists, at least I couldn't find one when I was looking for it. – Mattia Talpo May 11 '11 at 9:58

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.

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Yeah, thanks. I realised that yesterday (Biquard gives us a projective representation of the fundamental group which then is a representation of an extension of the same). – Vamsi May 13 '11 at 15:46