This paper is concerned with the computation of the inverse impulse response of a parametrized structural dynamics problem using reduced-order modeling and randomized excitations. A two-stages approach is proposed, involving the solution of both direct and inverse problems. In the first stage, the parametrized structural dynamics problem is formulated in the frequency domain, and solved using a reduced-order modeling approach. As a result, the parametric transfer function of the structure is obtained, and then readily transformed into a parametric direct impulse response (DIR). In the second stage, the parametric inverse impulse response (IIR) is computed. We use randomized excitations to generate synthetic samples inexpensively from the parametric DIR. Based on these, the parametric IIR is computed by minimizing the mean square error between the estimate and the samples. Most importantly, we show that the randomized excitations can be generated by sampling the frequency domain only. Hence, the parametric domain does not need to be sampled, which makes the computation of the parametric IIR very efficient.