LSM-tree-based key-value stores (KV stores) convert random-write requests to sequence-write ones to achieve high I/O performance. Meanwhile, compaction operations in KV stores update SSTables in forms of reorganizing low-level data components to high-level ones, thereby guaranteeing an orderly data layout in each component. Repeated writes caused by compaction (a.k.a. write amplification) impacts I/O bandwidth and overall system performance. Near-data processing (NDP) is one of the effective approaches to addressing this write-amplification issue. Most NDP-based techniques adopt synchronous parallel schemes to perform a compaction task on both the host and its NDP-enabled device. In synchronous parallel compaction schemes, the execution time of compaction is determined by a subsystem that has lower compaction performance coupled by under-utilized computing resources in a NDP framework. To solve this problem, we propose an asynchronous parallel scheme named PStore to improve the compaction performance in KV stores. In PStore, we designed a multi-tasks queue and three priority-based scheduling methods. PStore elects proper compaction tasks to be offloaded in host- and device-side compaction modules. Our proposed cross-leveled compaction mechanism mitigates write amplification induced by asynchronous compaction. PStore featured with the asynchronous compaction mechanism fully utilizes computing resources in both host- and device-side subsystems. Compared with the two popular synchronous compaction modes based on KV stores (TStore and LevelDB), our PStore immensely improves the throughput by up to a factor of 14 and 10.52 with an average of a factor of 2.09 and 1.73, respectively.