  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 $c=-10$.
