  In this paper, we generalize existing frameworks for
$\mathcal{H}_2\otimes\mathcal{L}_2$-optimal model order reduction to a broad
class of parametric linear time-invariant systems. To this end, we derive
first-order necessary ptimality conditions for a class of structured
reduced-order models, and then building on those, propose a
stability-preserving optimization-based method for computing locally
$\mathcal{H}_2\otimes\mathcal{L}_2$-optimal reduced-order models. We also make
a theoretical comparison to existing approaches in the literature, and in
numerical experiments, show how our new method, with reasonable computational
effort, produces stable optimized reduced-order models with significantly lower
approximation errors.
