We propose a statistical inference method for detecting change points in time-series of large panel data. The change points can have a general impact on different subsets of the panel. Our novel statistical perspective for high-dimensional change point detection combines selective inference and multiple testing. Our easy-to-use and computationally efficient procedure has two stages: First, LASSO regressions for each time-series screen a candidate set of change points. Second, we apply post-selection inference with a novel multiple testing adjustment to select the change points. Our method controls for the panel family-wise error rate with theoretical guarantees; hence guarding against p-hacking without the need for tuning parameters. In extensive simulations, our method outperforms leading benchmarks in terms of correct selections and false discovery. We have higher detection and make fewer Type I errors, leading to over 20% higher F1 classification scores.