  Based on the general formalism of parafermionic algebra and parasupersymmetry
proposed previously by us, we explicitly construct third-order parafermionic
algebra and multiplication law, and then realize third-order parasupersymmetric
quantum systems. We find some novel features in the third-order, namely, the
emergence of a fermionic degree of freedom and of a generalized parastatistics.
We show that for one-body cases the generalized Rubakov-Spiridonov model can be
constructed also in our framework and find that it admits a generalized 3-fold
superalgebra. We also find that a three-body system can have third-order
parasupersymmetry where three independent supersymmetries are folded. In both
cases, we also investigate the new concept of quasi-parasupersymmetry
introduced by us and find that those of order (3,3) are indeed realized under
less restrictive conditions than (ordinary) parasupersymmetric cases.
