  In the Fock representation, we construct matrix product states (MPS) for
one-dimensional gapped phases for $\mathbb{Z}_{p}$ parafermions. From the
analysis of irreducibility of MPS, we classify all possible gapped phases of
$\mathbb{Z}_{p}$ parafermions without extra symmetry other than
$\mathbb{Z}%_{p}$ charge symmetry, including topological phases, spontaneous
symmetry breaking phases and a trivial phase. For all phases, we find the
irreducible forms of local matrices of MPS, which span different kinds of
graded algebras. The topological phases are characterized by the non-trivial
simple $\mathbb{Z}_{p}$ graded algebras with the characteristic graded centers,
yielding the degeneracies of the full transfer matrix spectra uniquely. But the
spontaneous symmetry breaking phases correspond to the trivial semisimple
$\mathbb{Z}_{p/n}$ graded algebras, which can be further reduced to the trivial
simple $\mathbb{Z}_{p/n}$ graded algebras, where $n$ is the divisor of $p$. So
the present results deepen our understanding of topological phases in one
dimension from the viewpoints of MPS.
