Recent research and experimental findings, as well as technological development and commercialization efforts, suggest that even a modest amount of data can deliver superior signal modeling and reconstruction performance if sparsity is present and accounted for. Early sparsity-aware signal processing techniques have mostly targeted stationary signal analysis using offline algorithms for signal and image reconstruction from Fourier samples. On the other hand, sparsity-aware time-frequency tools for nonstationary signal analysis have recently received growing attention. In this context, sparse regression has offered a new paradigm for instantaneous frequency estimation, over classical time-frequency representations. Standard techniques for estimating model parameters from time series yield erroneous fits when, e.g., abrupt changes or outliers cause model mismatches. Accordingly, the need arises for basic research in robust processing of nonstationary parametric models that leverage sparsity to accomplish tasks such as tracking of signal variations, outlier rejection, robust parameter estimation, and change detection. This article aims at delineating the analytical background of sparsity-aware time-series analysis and introducing sparsity-aware robust and nonstationary parametric models to the signal processing readership, through readily appreciated applications in frequency-hopping (FH) communications and speech compression. Preliminary results strongly support the vision of seeking the right form of sparsity for the right application to enable sparsity-cognizant estimation of robust parametric models for nonstationary signal analysis.