If you take $r,s$ in $\mathbb{C}[u,v]$ homogeneous and of the same degree then Granville's bound seems correct.

In arxiv.org/abs/math/0411405 Steenbrink treats a similar problem in a special case.

Consider the period domain for the elliptic surfaces you are interested in. Then the set of points corresponding with elliptic surfaces with maximal Picard number is a countable set. Take now a one-dimensional family of elliptic surfaces, which, as a family of elliptic surfaces, is not isotrivial. Then this yields a curve in the period domain and this curve has to hit a point which corresponds to an elliptic surface with non-maximal Picard number