In this paper we reexamine the definition of parafermions and parabosons by means of Green's triple relations, and extend these relations by including a parity operator
P which is also determined by means of triple relations. As a consequence, we are dealing with new algebraic structures. It is shown that the algebra underlying a set of
n parafermions together with
P is the orthogonal Lie algebra
so(2n+2). The Fock spaces correspond to particular irreducible representations of
so(2n+2), and the action of
P in these spaces leads to interesting observations. Next, we show that the algebra underlying a set of
n parabosons together with
P is the orthosymplectic Lie superalgebra
osp(2∣2n). In this case, the Fock spaces correspond to certain irreducible infinite-dimensional representations of
osp(2∣2n). Both for parafermions and parabosons the spectrum of
P is closely related to the so-called order of statistics
p, introduced by Green.
Coordinate atypical representation of the orthosymplectic superalgebra osp(2/2) in a Hilbert superspace of square integrable functions constructed in a special way is given. The quantum nonrelativistic free particle Hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system. The supercoherent states, defined by means of a supergroup displacement operator, are explicitly constructed. These are the coordinate representation of the known atypical abstract super group OSp(2/2) coherent states. We interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement (in the appropriate