  Let $G$ be a simple algebraic group over an algebraically closed field $K$ of
characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$
that contain a given distinguished unipotent element $u$ of $G$. A result of
Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a
regular unipotent element, then $X$ cannot be contained in a proper parabolic
subgroup of $G$. We generalize their result and show that if $u$ has order $p$,
then except for two known examples which occur in the case $(G, p) = (C_2, 2)$,
the subgroup $X$ cannot be contained in a proper parabolic subgroup of $G$. In
the case where $u$ has order $> p$, we also present further examples arising
from indecomposable tilting modules with quasi-minuscule highest weight.
