[Present] Version nr. 3 (2005/06/01) is an extended version of Nr. 1 (2004/11/10), to which a section with the fixed determinant case was added. To appear in Memoirs of the AMS.Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles with fixed and non-fixed determinant, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles.Partially supported by Ministerio de Educación y Tecnología and Conselho de Reitores das Universidades Portuguesas through Acción Integrada Hispano-Lusa HP-2000-0015 (Spain) / E–30/03 (Portugal) and by the European Research Training Networks EDGE (Contract no. HPRN-CT-2000-00101) and EAGER (Contract no. HPRN-CT-2000-00099). Members of VBAC (Vector Bundles on Algebraic Curves). Second author partially supported by Centro de Matemática da Universidade do Porto, financed by FCT (Portugal) through the programmes POCTI and POSI of the QCA III (2000–2006) with European Community (FEDER) and national funds. First and third authors partially supported by Ministerio de Educación y Ciencia (Spain) through Project BFM2000-0024.Peer reviewed