Summary: ``The class of parity graphs, those in which the cardinality of every maximal independent subset of vertices has the same parity, contains the well-covered graphs and arose in connection with the PSPACE-complete game `generalized Kayles'. In 1983 [Ars Combin. {\bf 16} (1983), A, 189--198; MR0737090 (85j:90075)] we characterized parity graphs of girth 8 or more. This is extended to a characterization of the parity graphs of girth greater than 5. We deduce that these graphs can be recognized in polynomial time.''