  When the relative commutation relations between a set of m parafermions and n
parabosons are of ``relative parafermion type'', the underlying algebraic
structure is the classical orthosymplectic Lie superalgebra osp(2m+1|2n). The
relative commutation relations can also be chosen differently, of ``relative
paraboson type''. In this second case, the underlying algebraic structure is no
longer an ordinary Lie superalgebra, but a Z_2 x Z_2$-graded Lie superalgebra,
denoted here by pso(2m+1|2n). The identification of this new algebraic
structure was performed by Tolstoy, amongst others. In the present paper, we
investigate the subalgebra structure of pso(2m+1|2n). This allows us to study
the parastatistics Fock spaces for this new set of m+n para-operators, as they
correspond to lowest weight representations of pso(2m+1|2n). Our main result is
the construction of these Fock spaces, with a complete labeling of the basis
vectors and an explicit action of the para-operators on these basis vectors.
