Dynamic Graph Neural Networks (DGNNs) have been widely applied in various real-life applications, such as link prediction and pandemic forecast, to capture both static structural information and temporal characteristics from dynamic graphs. Combining both time-dependent and -independent components, DGNNs manifest substantial parallel computation and data reuse potentials, but suffer from severe memory access inefficiency and data transfer overhead under the canonical one-graph-at-a-time training pattern. To tackle these challenges, we propose PiPAD, a Pipelined and PArallel DGNN training framework for the end-to-end performance optimization on GPUs. From both algorithm and runtime level, PiPAD holistically reconstructs the overall training paradigm from the data organization to computation manner. Capable of processing multiple graph snapshots in parallel, PiPAD eliminates unnecessary data transmission and alleviates memory access inefficiency to improve the overall performance. Our evaluation across various datasets shows PiPAD achieves 1.22x - 9.57x speedup over the state-of-the-art DGNN frameworks on three representative models.