Since neural networks play an increasingly important role in critical sectors, explaining network predictions has become a key research topic. Counterfactual explanations can help to understand why classifier models decide for particular class assignments and, moreover, how the respective input samples would have to be modified such that the class prediction changes. Previous approaches mainly focus on image and tabular data. In this work we propose SPARCE, a generative adversarial network (GAN) architecture that generates SPARse Counterfactual Explanations for multivariate time series. Our approach provides a custom sparsity layer and regularizes the counterfactual loss function in terms of similarity, sparsity, and smoothness of trajectories. We evaluate our approach on real-world human motion datasets as well as a synthetic time series interpretability benchmark. Although we make significantly sparser modifications than other approaches, we achieve comparable or better performance on all metrics. Moreover, we demonstrate that our approach predominantly modifies salient time steps and features, leaving non-salient inputs untouched.