  We show that the single-mode parafermionic type systems possess
supersymmetry, which is based on the symmetry of characteristic functions of
the parafermions related to the generalized deformed oscillator of
Daskaloyannis et al. The supersymmetry is realized in both unbroken and
spontaneously broken phases. As in the case of parabosonic supersymmetry
observed recently by one of the authors, the form of the associated
superalgebra depends on the order of the parafermion and can be linear or
nonlinear in the Hamiltonian. The list of supersymmetric parafermionic systems
includes usual parafermions, finite-dimensional q-deformed oscillator,
q-deformed parafermionic oscillator and parafermionic oscillator with internal
$Z_2$ structure.
