I think that we can use the Theorem 5.3.3 of Carter's Simple book to find the $Z(P)$, where $P$ is a Sylow $p$-subgroup of a Chevalley group over a finite field of characteristic $p$ as will be shown below. But I am not sure! May someone helps me?
Theorem 5.3.3. Let $G=\mathbb{L}(K)$ be a Chevalley group, $U$ be the subgroup of $G$ generated by the root subgroups $X_r$ with $r\in > \Phi^+$ and $h(r)\geq m$. Then:
(i) U is nilpotent and $$U=U_1\supset U_2\supset\cdots \supset > U_h\supset1$$ is a central series for $U$, where $h$ is the greatest height of a root of $\mathbb{L}$.
(ii) ....
Since the above series is central, $U_h=Z(U)$. So it is enough to find $A=\{r\in \Phi^+\}$ with $h(r)=h$ and then find $Z(U)=<X_r:r\in A>$.