Let $G$ be a simple linear algebraic group defined over
$\mathbb{C}$ and $P$ a parabolic subgroup of it. Let $(M, E_P,
\omega)$ be a holomorphic parabolic geometry of type $G/P$
over a smooth complex projective variety $M$. We prove that
$(M, E_ , \omega)$ is holomorphically isomorphic to the
standard parabolic geometry $(G/P, G, \omega_0)$ whenever $M$
is rationally connected. We then show that this is indeed the
case if $M$ has Picard number one and contains a (possibly
singular) rational curve. This last result is a generalization
of the main result of [3], where we dealt with the case $G =
PGL(d, \mathbb{C})$, $G/P =
\mathbb{P}^{d-1}_{\mathbb{C}}$.