  To study exotic excitations described by parafermions in the possible spin
liquid states of SU($n$) spin systems, we introduce a parafermion parton
approach. The SU($n$) spin operators can be represented by clock and shift
matrices, which are shown to be the polynomials of parafermion operators in the
parafermion representation. We find that SU($n$) spins can be decomposed into
$n$ parafermion matrices of degree one. In this decomposition, the spin has a
$\{\bigotimes{\rm SU}(n)\}^{n-1}$ gauge symmetry. As an application, we study
the one-dimensional three-state clock model and generalized Kitaev model by a
mean-field theory, both of them have been proved to be related to parafermion
excitations. We find that with the symmetries of translations, $6$-fold
rotation and combination of parity and time reversal, there are $9$ types and
$102$ solutions for two-dimensional $Z_3$ parafermion spin liquids on the
honeycomb lattice. On the contrast, there are $9$ types and $36$ solutions if
both parity and time-reversal symmetries are present. Our results provide a
novel route for the systematic search of new types of spin liquids with exotic
anyon excitations.
