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 $S^{n-k}$, assuming that the degree of f is even in case $n=k$. 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 new parametrized Borsuk--Ulam and Bourgin--Yang--Volovikov type theorems.