This is too complicated, but it's the first approach that comes to mind. I take Conrad's ‘dynamical’ approach to parabolic subgroups, thinking instead of cocharacters defining them.

As you have observed, a unipotent element lies in the unipotent radical of some parabolic subgroup over the algebraic closure, hence is contracted by some cocharacter defined over the algebraic closure; so the closure of its orbit contains the identity. It follows from Corollary 4.3 of Kempf - Instability in invariant theory (MSN) that it is actually contracted by some rational cocharacter, hence lies in the unipotent radical of the corresponding rational parabolic subgroup.

This is too complicated, but it's the first approach that comes to mind. I take Conrad's ‘dynamical’ approach to parabolic subgroups, thinking instead of cocharacters defining them.

As you have observed, a nilpotent element lies in the Lie algebra of the unipotent radical of some parabolic subgroup over the algebraic closure, hence is strictly contracted by some cocharacter defined over the algebraic closure; so the closure of its orbit contains $0$. It follows from Corollary 4.3 of Kempf - Instability in invariant theory (MSN) that it is actually strictly contracted by some rational cocharacter, hence lies in the Lie algebra of the unipotent radical of the corresponding rational parabolic subgroup. (See also §2.5 of Adler and DeBacker - Some applications … (MSN).)

EDIT: The analogous result for unipotent elements is Lemma 8.3 of Borel and Tits - Groupes réductifs (MSN), and I suppose you could apply it to the group generated by your $x$.