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

Parafermionic chiral algebra Z_{3} with the dimension of the principal parafermion fields psi(z), psi^{+}(z), Delta_{psi}=8/3