What is a parametric excitation in structural dynamics?

It is commonly stated that a system is parametrically excited if the excitation appears as time-varying coefficients of the equations of motion. It is shown that this statement is contradicted in those structural problems with unconstrained motions whose excitation terms, either boundary forces or displacements, appear as inhomogeneities in the boundary conditions. Yet, these excitations, under pertinent conditions, may cause parametric Hill-type instabilities. It is only when suitable coordinate transformations are introduced or a constrained version of the motions is sought (e.g., via a Bubnov-Galerkin approach) that the parametric nature of the excitation is brought out explicitly and unambiguously.