The algebras $\mathfrak{osp}(1|2n)$ and $\mathfrak{so}(2n+1)$ describe the interactions of parabosonic and parafermionic particles, relevant in several areas of theoretical physics. The Lie superalgebra $\mathfrak{osp}(1|2n)$ can be generated by $n$ $p$-dimensional Dirac operators and vector variables. In a like manner the Lie algebra $\mathfrak{so}(2n+1)$ can be generated by $n$ $p$-dimensional Dirac operators and vector variables defined on Grassmannian instead of on Cartesian coordinates. We show how these realizations let us construct polynomial bases for the parabosonic and parafermionic Fock space representations of $\mathfrak{osp}(1|2n)$ and $\mathfrak{so}(2n+1)$. This is joint work with Hendrik De Bie and Joris Van der Jeugt.