Corollary 3.7 of the Borel-Tits 1971 Inventiones paper includes: if $K$ is a perfect field, $H$ any connected linear algebraic $K$-group, the maximal unipotent $K$-subgroups of $H$ are the unipotent radicals of minimal parabolic $K$-subgroup of $H$, and those are pairwise conjugate (by elements in $H(K)$). Here "unipotent" does not assume connected (Borel-Tits are very careful with this). In particular, in characteristic $p$, every finite $p$-subgroup of $H(K)$ is contained in the $K$-points of the unipotent radical of a minimal parabolic $K$-subgroup.