The unipotent radical $U$ of a maximal parabolic $P$ of a classical group is not always abelian - see Jim's answer to a specific example and the paper by Richardson, Rohrle and Steinberg for the general case. So in these cases $Z(U)$ is proper subgroup of $U$ that is normal in $P$, and you have a counterexample.

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.

Edit: As I mention in my comment above on Jim's answer, I don't know under what circumstances $Z(U)$ is minimal normal - again this would come down to studying the irreducibility of the action of the Levi on $Z(U)$.

The unipotent radical $U$ of a maximal parabolic $P$ of a classical group is not always abelian - see Jim's answer to a specific example and the paper by Richardson, Rohrle and Steinberg for the general case. So in these cases $Z(U)$ is proper subgroup of $U$ that is normal in $P$, and you have a counterexample.

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.

Edit: As I mention in my comment above on Jim's answer, I don't know under what circumstances $Z(U)$ is minimal normal - again this would come down to studying the irreducibility of the action of the Levi on $Z(U)$.

(I should add, in case anyone thinks I was cheeky, that I initially phrased this answer speculatively... and adjusted it in light of the counter-example given in Jim's answer.)

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.

The unipotent radical $U$ of a maximal parabolic $P$ of a classical group is not always abelian - see Jim's answer to a specific example and the paper by Richardson, Rohrle and Steinberg for the general case. So in these cases $Z(U)$ is proper subgroup of $U$ that is normal in $P$, and you have a counterexample.

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.

Edit: As I mention in my comment above on Jim's answer, I don't know under what circumstances $Z(U)$ is minimal normal - again this would come down to studying the irreducibility of the action of the Levi on $Z(U)$.