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Using the isomorphism between $N_{-3/2}(\\frak{sl}(3))$ and the logarithmic vertex algebra $\\mathcal{W}^{0} (2)_{A_2} $ from [2], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra $\\mathcal{W}(2,3)$ of central charge $c=-10$, and that $N_{-3/2}(\\frak{sl}(3))$ is a direct sum of irreducible $N_{-3/2}(\\frak{sl}(2))$-modules. As a byproduct, we prove certain conjectures about the vertex algebra $\\mathcal{W}^0(p)_{A_2}$. 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Using the isomorphism between $N_{-3/2}(\\frak{sl}(3))$ and the logarithmic vertex algebra $\\mathcal{W}^{0} (2)_{A_2} $ from [2], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra $\\mathcal{W}(2,3)$ of central charge $c=-10$, and that $N_{-3/2}(\\frak{sl}(3))$ is a direct sum of irreducible $N_{-3/2}(\\frak{sl}(2))$-modules. As a byproduct, we prove certain conjectures about the vertex algebra $\\mathcal{W}^0(p)_{A_2}$. 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