Given a (complex, smooth) irreducible representation $\pi$ of the general
linear group over a non-archimedean local field and an irreducible
supercuspidal representation $\sigma$ of a classical group, we show that the
(normalized) parabolic induction $\pi\rtimes\sigma$ is reducible if there
exists $\rho$ in the supercuspidal support of $\pi$ such that
$\rho\rtimes\sigma$ is reducible. In special cases we also give irreducibility
criteria for $\pi\rtimes\sigma$ when the above condition is not satisfied.