  In this paper we study the geometry of the moduli space of (non-strongly)
parabolic Higgs bundles over a Riemann surface with marked points. We show that
this space possesses a Poisson structure, extending the one on the dual of an
Atiyah algebroid over the moduli space of parabolic vector bundles. By
considering the case of full flags, we get a Grothendieck-Springer resolution
for all other flag types, in particular for the moduli spaces of twisted Higgs
bundles, as studied by Markman and Bottacin and used in the recent work of
Laumon-Ng\^o. We discuss the Hitchin system, and demonstrate that all these
moduli spaces are integrable systems in the Poisson sense.
