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Change-point (CP) VAR models face a dimensionality curse due to the proliferation of parameters that arises when new breaks are detected. We introduce the Sparse CP-VAR model which determines which parameters truly vary when a break is detected. By doing so, the number of new parameters to be estimated at each regime is drastically reduced and the break dynamics becomes easier to be interpreted. The Sparse CP-VAR model disentangles the dynamics of the mean parameters and the covariance matrix. The former uses CP dynamics with shrinkage prior distributions, while the latter is driven by an infinite hidden Markov framework. An extensive simulation study is carried out to compare our approach with existing ones. We provide applications to financial and macroeconomic systems. It turns out that many off-diagonal VAR parameters are zero for the entire sample period and that most break activity is in the covariance matrix. We show that this has important consequences for portfolio optimization, in particular when future instabilities are included in the predictive densities. Forecasting-wise, the Sparse CP-VAR model compares favorably to several time-varying parameter models in terms of density and point forecast metrics.