It is a general fact that if $P$ is a parabolic $k$-subgroup of an arbitrary connected reductive group $G$ over an arbitrary field $k$ then $U := \mathscr{R}_u(P)$ has a canonical $P$-equivariant filtration $$U = U_0 \supset U_1 \supset \dots \supset U_m \supset U_{m+1} = \{1\}$$ by smooth connected $k$-subgroups such that each $U_i/U_{i+1}$ is a vector group that admits a unique $P$-equivariant linear structure. (This can be defined over $\mathbf{Z}$ as well when working with "Chevalley groups", so it is truly characteristic-free). In particular, the final stage of this canonical filtration is a vector group that is a normal $k$-subgroup of $P$. This must be well-known in the classical case, but the only reference I know in the literature is SGA3, Exp. XXVI, 2.1.

If $G$ is split and $T$ is a split maximal $k$-torus in $P$ then the resulting linear representation of $T$ on each $U_i/U_{i+1}$ is a direct sum of 1-dimensional weight spaces given by certain roots in $\Phi(G,T)$. I expect that when the root system is moreover irreducible then each $(U_i/U_{i+1})(k) = U_i(k)/U_{i+1}(k)$ is irreducible as a $P(k)$-representation, apart from perhaps some special cases with low rank and/or $k$ with size 2 or 3. In particular, the final stage $U_m(k)$ would then be minimal as a normal subgroup of $P(k)$. Maybe this is sufficient for your purposes? (No motivation was given for the posted question, beyond perhaps curiosity.)

Since you use the notation "PSL" which is a perennial source of confusion (do you mean ${\rm{SL}}_n(k)/\mu_n(k)$ or ${\rm{PSL}}_n(k)$ where ${\rm{PSL}}_n$ is meant as the quotient ${\rm{SL}}_n/\mu_n$ in the sense of algebraic groups?), I should note that although such confusion creates headaches with the tori, it does not with the "unipotent radicals", so the above should be applicable to various possible meanings of the phrase "finite classical group", but (as always) be careful.

It is a general fact that if $P$ is a parabolic $k$-subgroup of an arbitrary connected reductive group $G$ over an arbitrary field $k$ then $U := \mathscr{R}_u(P)$ has a canonical $P$-equivariant filtration $$U = U_0 \supset U_1 \supset \dots \supset U_m \supset U_{m+1} = \{1\}$$ by smooth connected $k$-subgroups such that each $U_i/U_{i+1}$ is a vector group that admits a unique $P$-equivariant linear structure. (This can be defined over $\mathbf{Z}$ as well when working with "Chevalley groups", so it is truly characteristic-free). In particular, the final stage of this canonical filtration is a vector group that is a normal $k$-subgroup of $P$. This must be well-known in the classical case, but the only reference I know in the literature is SGA3, Exp. XXVI, 2.1.

If $G$ is split semisimple of adjoint type and $T$ is a split maximal $k$-torus in $P$ then the resulting linear representation of $T$ on each $U_i/U_{i+1}$ is a direct sum of 1-dimensional weight spaces given by certain roots in $\Phi(G,T)$. I expect that when the root system is moreover irreducible then each $(U_i/U_{i+1})(k) = U_i(k)/U_{i+1}(k)$ is irreducible as a $k$-linear $P(k)$-representation, apart from perhaps some special cases with low rank and/or $k$ with size 2 or 3. In particular, the final stage $U_m(k)$ would then be minimal as a normal subgroup of $P(k)$ (since there are no divisible roots in the character lattice due to $G$ being of adjoint type). Maybe this is sufficient for your purposes? (No motivation was given for the posted question, beyond perhaps curiosity.)

Since you use the notation "PSL" which is a perennial source of confusion (do you mean ${\rm{SL}}_n(k)/\mu_n(k)$ or ${\rm{PSL}}_n(k)$ where ${\rm{PSL}}_n$ is meant as the quotient ${\rm{SL}}_n/\mu_n$ in the sense of algebraic groups?), I should note that although such confusion creates headaches with the tori, it does not with the "unipotent radicals", so the above should be applicable to various possible meanings of the phrase "finite classical group", but (as always) be careful.