This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:

Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, are determined by faces $c$ of the positive Weyl chamber. Interiors of *some* of these faces contain coroots of the group $G$. Is there a name for the class of parabolic subgroups $P$ and the corresponding partial flag manifolds $G/P$ coming from such faces?

(I do not know the name even in the special case when such $P$ is maximal parabolic. Such maximal parabolics exist except for the type $A$ and $D_3$ groups: Most of the time they correspond to the highest (co)root.) My tentative name for $P$'s and $G/P$'s is "root type parabolic subgroups and partial flag manifolds". In a paper I am cowriting, such subgroups and manifolds behave better than other parabolics.