I don't believe that the unipotent radical $U$ of a maximal parabolic $P$ of a classical group is always abelian - see, for instance, the paper by Richardson, Rohrle and Steinberg. So in these cases $Z(U)$ is proper subgroup of $U$ that is normal in $P$.
In the cases where $U$ is abelian, one needs to check whether the natural action of a Levi subgroup of $P$ on the unipotent radical $U$ is irreducible. There are lots of sources for this sort of thing, for instance Volume 3 of the series by Gorenstein, Lyons and Solomon.