We develop a general framework for sparse and adaptive interpolation of histograms/measures from the Wasserstein space of probability measures. The strategy relies on approximation of measures with Wasserstein barycenters. The optimal performance of the approach in terms of approximation error is characterized by a notion of best $n$-term barycentric approximation which we introduce in the talk. This best approximation is the minimizer of a highly non-convex, bi-level optimization problem, and we develop algorithmic strategies for practical numerical computation. We next leverage this approximation tool in order to build interpolation strategies to address structured prediction problems where the family of measures to approximate presents common general structural features. To illustrate the potential of the method, we focus on Model Order Reduction (MOR) of parametrized PDEs. The whole methodology is computationally feasible thanks to state of the art entropy regularized Sinkhorn algorithms from the field of numerical optimal transport.