  We propose an elegant formulation of parafermionic algebra and
parasupersymmetry of arbitrary order in quantum many-body systems without
recourse to any specific matrix representation of parafermionic operators and
any kind of deformed algebra. Within our formulation, we show generically that
every parasupersymmetric quantum system of order p consists of N-fold
supersymmetric pairs with N<p or N=p and thus has weak quasi-solvability and
isospectral property. We also propose a new type of non-linear supersymmetries,
called quasi-parasupersymmetry, which is less restrictive than
parasupersymmetry and is different from N-fold supersymmetry even in one-body
systems though the conserved charges are represented by higher-order linear
differential operators. To illustrate how our formulation works, we construct
second-order parafermionic algebra and three simple examples of
parasupersymmetric quantum systems of order 2, one is essentially equivalent to
the one-body Rubakov-Spiridonov type and the others are two-body systems in
which two supersymmetries are folded. In particular, we show that the first
model admits a generalized 2-fold superalgebra.
