Using Bony's paramultiplication we improve a result obtained in [J. Math. Pures Appl. 84 (2005), no. 4, 471--491] for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ is replaced by ${\mathcal C}^{1,\varepsilon}$, with $\varepsilon>0
