By studying the Galois action on etale cohomology groups arising from families of elliptic curves, we will prove several new cases of the regular Inverse Galois Problem. In particular, we will explain why each of the simple groups $\mathrm{PSp}_4(\mathbf{F}_p)$ occur as the Galois group of a regular extension of the function field $\mathbf{Q}(t)$. The key ingredients will be a big monodromy result along with some known cases of the Birch and Swinnerton-Dyer conjecture.