  We extend to characteristic $2$ and $3$ the classification of projective
homogeneous varieties of Picard group isomorphic to $\mathbf{Z}$, corresponding
to parabolic subgroup schemes with maximal reduced subgroup. The latter are all
obtained as product of a maximal reduced parabolic with the kernel of a purely
inseparable isogeny. This fails in type $G_2$ and characteristic $2$, for which
we exhibit an explicit counterexample and show it is the only one, thus
completing the classification. We then construct new examples of projective
homogeneous varieties of Picard rank two.
