We present a new class of parallel and distributed audio concealment (PDAC) algorithms which recover lost audio packets at the receiver to fight against channel impairment. The main contribution of this work is the proposal of using nonlocal sparse representations to characterize the prior constraint of undamaged audio. When combined with observation constraint, we obtain an alternating projection based audio concealment algorithm which recovers missing data in a parallel and distributed fashion. We also present two extensions of PDAC for more challenging situations: expectation-maximization PDAC (EM-PDAC) to handle consecutive packet loss and filter-bank PDAC (FB-PDAC) to repair complex music signals. Excellent preliminary experimental results are reported for a wide range of audio materials and loss conditions.