  In the Fock representation, we propose a framework to construct the
generalized matrix product states (MPS) for topological phases with $\mathbb{
Z}_{p}$ parafermions. Unlike the $\mathbb{Z}_{2}$ Majorana fermions, the $%
\mathbb{Z}_{p}$ parafermions form intrinsically interacting systems. Here we
explicitly construct two topologically distinct classes of irreducible $%
\mathbb{Z}_{3}$ parafermionic MPS wave functions, characterized by one or two
parafermionic zero modes at each end of an open chain. Their corresponding
parent Hamiltonians are found as the fixed point models of the single
$\mathbb{Z}_{3}$ parafermion chain and two-coupled parafermion chains with
$\mathbb{Z}_{3}\times \mathbb{Z}_{3}$ symmetry. Our results thus pave the road
to investigate all possible topological phases with $\mathbb{Z}_{p}$
parafermions within the matrix product representation in one dimension.
