Speculative decoding enhances the inference efficiency of large language models (LLMs) by generating drafts using a small draft language model (DLM) and verifying them in batches with a large target language model (TLM). However, adaptive drafting inference on a mobile single-NPU-PIM system faces idle overhead in traditional operator-level synchronous execution and wasted computation in asynchronous execution due to fluctuations in draft length. This paper introduces AHASD, a task-level asynchronous mobile NPU-PIM heterogeneous architecture for speculative decoding. Notably, AHASD achieves parallel drafting on the PIM and verification on a single NPU through task-level DLM-TLM decoupling and specifically, it incorporates Entropy-History-Aware Drafting Control and Time-Aware Pre-Verification Control to dynamically manage adaptive drafting algorithm execution and pre-verification timing, suppressing invalid drafting based on low-confidence drafts. Additionally, AHASD integrates Attention Algorithm Units and Gated Task Scheduling Units within LPDDR5-PIM to enable attention link localization and sub-microsecond task switching on the PIM side. Experimental results for different LLMs and adaptive drafting algorithms show that AHASD achieves up to 4.2$\times$ in throughput and 5.6$\times$ in energy efficiency improvements over a GPU-only baseline, and 1.5$\times$ in throughput and 1.24$\times$ in energy efficiency gains over the state-of-the-art GPU+PIM baseline, with hardware overhead below 3% of the DRAM area.