Equivalence between algebraic structures generated by parastatisticstriple relations of Green (1953) and Greenberg -- Messiah (1965), and certain orthosymplectic $mathbb{Z}_2times mathbb{Z}_2$-graded Lie superalgebras is found explicitly. Moreover, it is shown that such superalgebras give more complex para-Fermi and para-Bose systems then ones of Green -- Greenberg -- Messiah.
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