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. We then construct a new class of homogeneous varieties of Picard rank two.