We study parafermion vertex algebras $N_{ -3/2}(\frak{ sl}(2))$ and $N_{ -3/2}(\frak{ sl}(3))$. 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 $\mathcal{ W}^0(p)_{ A_2}$. We also obtain a vertex-algebraic proof of the irreducibility of a family of $\mathcal W(2,3)_{ c}$ modules at $c=-10$.
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