  A thorough analysis of the general features of $(p=2)$ parasupersymmetric
quantum mechanics is presented. It is shown that for both Rubakov--Spiridonov
and Beckers--Debergh formulations of $(p=2)$-parasupersymmetric quantum
mechanics, the degeneracy structure of the energy spectrum can be derived using
the defining parasuperalgebras. Thus the results of the present article is
independent of the details of the Hamiltonian. In fact, they are valid for
arbitrary systems based on arbitrary dimensional coordinate manifolds. In
particular, the Rubakov--Spiridonov (R-S) and Beckers--Debergh (B-D) systems
possess identical degeneracy structures. For a subclass of R-S (alternatively
B-D) systems, a new topological invariant is introduced. This is a counterpart
of the Witten index of the supersymmetric quantum mechanics.
