  We consider the moduli space of stable parabolic Higgs bundles of rank $r$
and fixed determinant, and having full flag quasi-parabolic structures over an
arbitrary parabolic divisor on a smooth complex projective curve $X$ of genus
$g$, with $g\,\geq\, 2$. The group $\Gamma$ of $r$-torsion points of the
Jacobian of $X$ acts on this moduli space. We describe the connected components
of the various fixed point loci of this moduli under non-trivial elements from
$\Gamma$. When the Higgs field is zero, or in other words when we restrict
ourselves to the moduli of stable parabolic bundles, we also compute the
orbifold Euler characteristic of the corresponding global quotient orbifold. We
also describe the Chen--Ruan cohomology groups of this orbifold under certain
conditions on the rank and degree, and describe the Chen--Ruan product
structure in special cases.
