On parafermion vertex algebras of $\mathfrak{s}\mathfrak{l}(2{)}_{-3\mathrm{/}2}$ and $\mathfrak{s}\mathfrak{l}(3{)}_{-3\mathrm{/}2}$

We study parafermion vertex algebras $N_{-3\mathrm{/}2}(\mathfrak{s}\mathfrak{l}(2))$ and $N_{-3\mathrm{/}2}(\mathfrak{s}\mathfrak{l}(3))$. Using the isomorphism between $N_{-3\mathrm{/}2}(\mathfrak{s}\mathfrak{l}(3))$ 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 $N_{-3\mathrm{/}2}(\mathfrak{s}\mathfrak{l}(3))$ is a direct sum of irreducible $N_{-3\mathrm{/}2}(\mathfrak{s}\mathfrak{l}(2))$-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 $c=-10$.