  We consider a family of boundary integral operators supported on a collection
of parametrically defined bounded Lipschitz boundaries. Consequently, the
boundary integral operators themselves also depend on the parametric variables,
thus leading to a parameter-to-operator map. The main result of this article is
to establish the analytic or holomorphic dependence of said boundary integral
operators upon the parametric variables, i.e., of the parameter-to-operator
map. As a direct consequence we also establish holomorphic dependence of
solutions to boundary integral equations, i.e.,~holomorphy of the
parameter-to-solution map. To this end, we construct a holomorphic extension to
complex-valued boundary deformations and investigate the \emph{complex}
Fr\'echet differentiability of boundary integral operators with respect to each
parametric variable. The established parametric holomorphy results have been
identified as a key property to overcome the so-called curse of dimensionality
in the approximation of parametric maps with distributed, high-dimensional
inputs.
  To demonstrate the applicability of the derived results, we consider as a
concrete example the sound-soft Helmholtz acoustic scattering problem and its
frequency-robust boundary integral formulations. For this particular
application, we explore the consequences of our results in reduced order
modelling, Bayesian shape inversion, and the construction of efficient
surrogates using artificial neural networks.
