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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 areinfinite 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 abyproduct, we prove certain conjectures about the vertex algebra$\mathcal{W}^0(p)_{A_2}$. We also obtain a vertex-algebraic proof of theirreducibility of a family of $\mathcal W(2,3)_{c}$ modules at $c=-10$.

On parafermion vertex algebras of $\frak{sl}(2)_{-3/2}$ and $\frak{sl}(3)_{-3/2}$