  The ($p=2$) parabose-parafermi supersymmetry is studied in general terms. It
is shown that the algebraic structure of the ($p=2$) parastatistical dynamical
variables allows for (symmetry) transformations which mix the parabose and
parafermi coordinate variables. The example of a simple parabose-parafermi
oscillator is discussed and its symmetries investigated. It turns out that this
oscillator possesses two parabose- -parafermi supersymmetries. The combined set
of generators of the symmetries forms the algebra of supersymmetric quantum
mechanics supplemented with an additional central charge. In this sense there
is no relation between the parabose-parafermi supersymmetry and the
parasupersymmetric quantum mechanics. A precise definition of a quantum system
involving this type of parabose- parafermi supersymmetry is offered, thus
introducing ($p=2$) Supersymmetric Paraquantum Mechanics. The spectrum
degeneracy structure of general ($p=2$) supersymmetric paraquantum mechanics is
analyzed in detail. The energy eigenvalues and eigenvectors for the
parabose-parafermi oscillator are then obtained explicitly. The latter confirms
the validity of the results obtained for general supersymmetric paraquantum
mechanics.
