MPS representation to the one-dimensional parafermion systems. Recently, the fermionic MPS have been successfully constructed by using the language of super vector space, and all possible topological phases with additional symmetries in terms of Majorana fermions have been classified within the matrix product representation[@Bultinck2017; @Kapustin2016]. By generalizing the concepts of fermion parity and associated Fock space, the present authors have proposed a general framework to construct the MPS of $\mathbb{Z}_{3}$ parafermions in the Fock representation, and the corresponding parent Hamiltonians have been also derived[@XuZhang]. Therefore, the road has been cleared to classify all possible $\mathbb{Z}_{p}$ parafermion gapped phases within the framework of the MPS.

In this paper, we first review the Fock space of parafermions and then construct the parafermionic MPS. From the analysis of irreducibility of these parafermionic MPS, we provide the complete classification of all possible gapped phases without extra symmetry, compared to the previous classification based on the edge fractionalization[@Pollmann2013; @Quella]. More importantly, we find that the various irreducible forms for local matrices of MPS spanned different kinds of graded algebras characterize distinct parafermionic gapped phases. The local matrices of MPS describing topological phases span the non-trivial simple $\mathbb{Z}_{p}$ graded algebras with characteristic centers, resulting in the 