  We develop an optimization-based algorithm for parametric model order
reduction (PMOR) of linear time-invariant dynamical systems. Our method aims at
minimizing the $\mathcal{H}_\infty \otimes \mathcal{L}_\infty$ approximation
error in the frequency and parameter domain by an optimization of the reduced
order model (ROM) matrices. State-of-the-art PMOR methods often compute several
nonparametric ROMs for different parameter samples, which are then combined to
a single parametric ROM. However, these parametric ROMs can have a low accuracy
between the utilized sample points. In contrast, our optimization-based PMOR
method minimizes the approximation error across the entire parameter domain.
Moreover, due to our flexible approach of optimizing the system matrices
directly, we can enforce favorable features such as a port-Hamiltonian
structure in our ROMs across the entire parameter domain.
  Our method is an extension of the recently developed SOBMOR-algorithm to
parametric systems. We extend both the ROM parameterization and the adaptive
sampling procedure to the parametric case. Several numerical examples
demonstrate the effectiveness and high accuracy of our method in a comparison
with other PMOR methods.
