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 Sn−k. We present a parametrized generalization. For the proof we introduce a Fadell-Husseini type ideal-valued index of G-bundles that is quite computable in our situation and we obtain two new parametrized Borsuk-Ulam type theorems.