  In this paper we characterize the rank two vector bundles on $\mathbb{P}^2$
which are invariant under the actions of the parabolic subgroups
$G_p:=\mathrm{Stab}_p(\mathrm{PGL}(3))$ fixing a point in the projective plane,
$G_L:=\mathrm{Stab}_L(\mathrm{PGL}(3))$ fixing a line, and when $p\in L$, the
Borel subgroup $\mathbf{B} = G_p \cap G_L$ of $\mathrm{PGL}(3)$. Moreover, we
prove that the geometrical configuration of the jumping locus induced by the
invariance does not, on the other hand, characterize the invariance itself.
Indeed, we find infinite families that are almost uniform but not almost
homogeneous.
