Gromov, Memarian, and Karasev--Volovikov proved that any map f from an n-sphere to a k-manifold (n>=k) has a preimage f^{-1}(z) whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator S^{n-k} (if n=k we need that the degree of f is even). We present a parametrized generalization. For the proof we introduce a Fadell--Husseini index type theorem for G-bundles, and we obtain two new parametrized Borsuk--Ulam type theorems.