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](https://www.jstor.org/stable/1971168) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=506989)) that it is actually contracted by some *rational* cocharacter, hence lies in the unipotent radical of the corresponding *rational* parabolic subgroup.