  We analyze, and prove, the associativity of the new Z_{3} parafermionic
chiral algebra which has been announced some time ago, with principal
parafermionic fields having the conformal dimension \Delta_{\psi}=8/3. In doing
so we have developed a new method for analyzing the associativity of a given
chiral algebra of parafermionic type, the method which might be of a more
general significance than a particular conformal field theory studied in detail
in this paper. Still, even in the context of our particular chiral algebra, of
Z_{3} parafermions with \Delta_{\psi}=8/3, the new method allowed us to give a
proof of associativity which we consider to be complete.
