  The general features of the degeneracy structure of ($p=2$)
parasupersymmetric quantum mechanics are employed to yield a classification
scheme for the form of the parasupersymmetric Hamiltonians. The method is
applied to parasupersymmetric systems whose Hamiltonian is the square root of a
forth order polynomial in the generators of the parasupersymmetry. These
systems are interesting to study for they lead to the introduction of a set of
topological invariants very similar to the Witten indices of ordinary
supersymmetric quantum mechanics. The topological invariants associated with
parasupersymmetry are shown to be related to a pair of Fredholm operators
satisfying two compatibility conditions. An explicit algebraic expression for
the topological invariants of a class of parasupersymmetric systems is
provided.
