With the growing popularity of shared resources, large volumes of complex data of different types are collected automatically. Traditional data mining algorithms generally have problems and challenges including huge memory cost, low processing speed, and inadequate hard disk space. As a fundamental task of data mining, sequential pattern mining (SPM) is used in a wide variety of real-life applications. However, it is more complex and challenging than other pattern mining tasks, i.e., frequent itemset mining and association rule mining, and also suffers from the above challenges when handling the large-scale data. To solve these problems, mining sequential patterns in a parallel or distributed computing environment has emerged as an important issue with many applications. In this article, an in-depth survey of the current status of parallel SPM (PSPM) is investigated and provided, including detailed categorization of traditional serial SPM approaches, and state-of-the art PSPM. We review the related work of PSPM in details including partition-based algorithms for PSPM, apriori-based PSPM, pattern-growth-based PSPM, and hybrid algorithms for PSPM, and provide deep description (i.e., characteristics, advantages, disadvantages, and summarization) of these parallel approaches of PSPM. Some advanced topics for PSPM, including parallel quantitative/weighted/utility SPM, PSPM from uncertain data and stream data, hardware acceleration for PSPM, are further reviewed in details. Besides, we review and provide some well-known open-source software of PSPM. Finally, we summarize some challenges and opportunities of PSPM in the big data era.