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?
EDIT: As Jim Humphreys points out below, there is one pathological case, where $G_1\cap P_2$ might be all of $G_1$. In which case, all parabolic sub-groups of $G_1$ will be contained in $P_2$! This is the case, for example, when $G_1$ is a Levi sub-group of $G_2$. But Angelo's answer below shows that, excluding this possibility, the answer to my third question is 'yes'.