Suppose that we have a closed embedding $G_1\hookrightarrow G_2$ of reductive groups (say over $\mathbb{Q}$), and suppose that we have a maximal parabolic sub-group $P_2\subset G_2$, and a minimal parabolic $P_1\subset G_1$. Is it possible to have two different maximal parabolic sub-groups of $G_1$ contained in $P_2$ and containing $P_1$?

Actually, is it even possible that there are two different maximal parabolics of $G_1$ contained in $P_2$?

Even more optimistically, if there is a maximal parabolic sub-group of $G_1$ contained in $G_1\cap P_2$, does that make $G_1\cap P_2$ a parabolic sub-group and thus equal to the maximal parabolic it contains?