Sequential pattern mining can be applied to various fields such as disease prediction and stock analysis. Many algorithms have been proposed for sequential pattern mining, together with acceleration methods. In this paper, we show that a heterogeneous platform with CPU and GPU is more suitable for sequential pattern mining than traditional CPU-based approaches since the support counting process is inherently succinct and repetitive. Therefore, we propose the PArallel SequenTial pAttern mining algorithm, referred to as PASTA, to accelerate sequential pattern mining by combining the merits of CPU and GPU computing. Explicitly, PASTA adopts the vertical bitmap representation of database to exploits the GPU parallelism. In addition, a pipeline strategy is proposed to ensure that both CPU and GPU on the heterogeneous platform operate concurrently to fully utilize the computing power of the platform. Furthermore, we develop a swapping scheme to mitigate the limited memory problem of the GPU hardware without decreasing the performance. Finally, comprehensive experiments are conducted to analyze PASTA with different baselines. The experiments show that PASTA outperforms the state-of-the-art algorithms by orders of magnitude on both real and synthetic datasets.