There are several questions being asked (and an unexplained reference to a field of definition), but the answer to at least one of them is no: Take $G_1 = G_2 = \mathrm{SL}_3(\mathbb{C})$, with a given minimal = maximal parabolic subgroup involving a single simple root subgroup relative to some choice of positive roots; this parabolic clearly won't lie in two distinct maximal ones.   (I'm assuming "minimal" excludes a Borel subgroup and "maximal" means proper, though
a reductive group might be just a torus.)

So it's a good idea to separate out the more precise question you have in mind and specify whether a field of definition is really involved for the various groups and subgroups involved.