For a semisimple real Lie group G, we study topological properties of moduli spaces of parabolic G-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichmüller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for parabolic maximal G-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit how one can provide an alternative explanation of fundamental results on counting components in the absence of a parabolic structure. These topological invariants may be found useful in the study of the geometric Langlands program in the case of tame ramification.

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For a connected complex reductive group G, we introduce a notion of stability for parahoric G θ -Higgs bundles over a smooth algebraic curve X, where G θ is a parahoric group scheme over X. In the case when the group G is the general linear group GLn, we show that the stability condition of a parahoric torsor reduces to stability of a parabolic bundle. A correspondence between semistable tame parahoric G θ -Higgs bundles and semistable tame equivariant G-Higgs bundles allows us to construct the moduli space explicitly. This moduli space is shown to be equipped with a Poisson structure.