  We study deformations of smooth mod $p$ representations (and their duals) of
a $p$-adic reductive group $G$. Under some mild genericity condition, we prove
that parabolic induction with respect to a parabolic subgroup $P=LN$ defines an
isomorphism between the universal deformation rings of a supersingular
representation $\bar{\sigma}$ of $L$ and of its parabolic induction
$\bar{\pi}$. As a consequence, we show that every Banach lift of $\bar{\pi}$ is
induced from a unique Banach lift of $\bar{\sigma}$.
