Presumably the argument ilya was driving at is this: If one had a surjective map of group schemes G -> H, then consider the preimage of the nilradical of H. This is an algebraic group with a surjective map to a unipotent group. Since there are no group homomorphisms from reductive groups to unipotent ones, the nilradical of this preimage (which is contained in the nilradical of G) must surject onto the nilradical of H. So if the former is trivial, so is the latter.

Presumably the argument ilya was driving at is this: If one had a surjective map of group schemes $G \to H$, then consider the preimage of the nilradical of $H$. This is an algebraic group with a surjective map to a unipotent group. Since there are no group homomorphisms from reductive groups to unipotent ones, the nilradical of this preimage (which is contained in the nilradical of $G$) must surject onto the nilradical of $H$. So if the former is trivial, so is the latter.