The classical Gromov width measures the largest symplectic ball embeddable
into a symplectic manifold; inspired by the symplectic camel problem, we
generalize this to ask how large a symplectic ball can be embedded as a family
over a parameter space $N$. Given a smooth map $f: N \to \Omega$, where
$\Omega$ is a symplectic manifold, we define the parametric Gromov width
$\mathrm{Gr}(f,\Omega)$ as the supremum of capacities $a$ for which there
exists $F: N \times B(a) \to \Omega$ with $F(\eta, 0) = f(\eta)$ and which
restricts to a symplectic embedding on each ball $\{ \eta \} \times B(a)$,
where $B(a) \subset \mathbb{C}^n$ is the closed ball of capacity $a$. For
Liouville domains $\Omega$, we establish upper bounds on
$\mathrm{Gr}(f,\Omega)$ using the Floer cohomology persistence module
associated to $\Omega$. Specializing to fiberwise starshaped domains in the
cotangent bundle $T^*M$, we derive computable bounds via filtered string
topology. Specific examples of $\Omega$ — including disk cotangent bundles of
thin ellipsoids, open books, and tori — demonstrate our bounds, and reveal
constraints on parameterized symplectic embeddings beyond the classical Gromov
width.

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