  Based on the recently established parafermionic matrix product states, we
study the classification of one-dimensional gapped phases of parafermions with
the time reversal (TR) symmetry satisfying $T^{2}=1$. Without extra symmetry,
it has been found that $\mathbb{Z}_{p}$ parafermionic gapped phases can be
classified as topological phases, spontaneous symmetry breaking (SSB) phases,
and a trivial phase, which are uniquely labelled by the divisors $n$ of $p$. In
the presence of TR symmetry, however, the enriched classification is
characterized by three indices $n$, $\kappa $ and $\mu $, where $\kappa \in
\mathbb{Z}_{2}$ denotes the linear or projective TR actions on the edges, and
$\mu \in \mathbb{Z}_{2}$ indicates the commutation relations between the TR and
(fractionalized) charge operator. For the $\mathbb{Z}_{r}$ symmetric
parafermionic ground states, where $r=p$ for trivial or topological phases, and
$r=p/n$ for SSB phases, the original gapped phases with odd $r$ are divided
into two phases, while those phases with even $r$ are further separated into
four phases. The gapped parafermionic phases with the TR symmetry include the
symmetry protected topological phases, symmetry enriched topological phases,
and the SSB coexisting symmetry protected topological phases. From analyzing
the structures and symmetries of their reduced density matrices of those
resulting topological phases, we can obtain the topological protected
degeneracies of their entanglement spectra.
