Abstract:Parabolic U(p,q)-Higgs bundles over a compact Riemann surface with a finite set of marked points are objects that correspond to representations of the fundamental group of the surface without the marked points, with fixed holonomy classes around them. We count the number of connected components of the moduli space of parabolic U(p,q)-Higgs bundles. The main strategy is to use Bott-Morse theoretic techniques using the L2-norm of the Higgs field as Morse function. The connectedness properties of our moduli space reduce to the connectedness of certain moduli spaces of parabolic triples, so we study also the irreducibility of the moduli spaces of parabolic triples.