Sparsely-activated Mixture-of-Expert (MoE) layers have found practical applications in enlarging the model size of large-scale foundation models, with only a sub-linear increase in computation demands. Despite the wide adoption of hybrid parallel paradigms like model parallelism, expert parallelism, and expert-sharding parallelism (i.e., MP+EP+ESP) to support MoE model training on GPU clusters, the training efficiency is hindered by communication costs introduced by these parallel paradigms. To address this limitation, we propose Parm, a system that accelerates MP+EP+ESP training by designing two dedicated schedules for placing communication tasks. The proposed schedules eliminate redundant computations and communications and enable overlaps between intra-node and inter-node communications, ultimately reducing the overall training time. As the two schedules are not mutually exclusive, we provide comprehensive theoretical analyses and derive an automatic and accurate solution to determine which schedule should be applied in different scenarios. Experimental results on an 8-GPU server and a 32-GPU cluster demonstrate that Parm outperforms the state-of-the-art MoE training system, DeepSpeed-MoE, achieving 1.13$\times$ to 5.77$\times$ speedup on 1296 manually configured MoE layers and approximately 3$\times$ improvement on two real-world MoE models based on BERT and GPT-2.