Reasoning about asynchronous plans is challenging since it requires sequential and parallel planning to optimize time costs. Can large language models (LLMs) succeed at this task? Here, we present the first large-scale study investigating this question. We find a representative set of closed and open-source LLMs, including GPT-4 and LLaMA-2, behave poorly when not supplied with illustrations about the task-solving process in our benchmark AsyncHow. We propose a novel technique Plan Like a Graph (PLaG) that combines graphs with natural language prompts and achieves state-of-the-art performances. We show that although our proposed method can boost model performance, LLMs still suffer from drastic performance degradation when task complexity increases, highlighting the limits of utilizing LLMs for simulating digital devices. We see our study as an exciting step towards using LLMs as efficient autonomous agents.