The development of quantum machine learning (QML) has received a lot of interest recently thanks to developments in both quantum computing (QC) and machine learning (ML). One of the ML paradigms that can be utilized to address challenging sequential decision-making issues is reinforcement learning (RL). It has been demonstrated that classical RL can successfully complete many difficult tasks. A leading method of building quantum RL agents relies on the variational quantum circuits (VQC). However, training QRL algorithms with VQCs requires significant amount of computational resources. This issue hurdles the exploration of various QRL applications. In this paper, we approach this challenge through asynchronous training QRL agents. Specifically, we choose the asynchronous training of advantage actor-critic variational quantum policies. We demonstrate the results via numerical simulations that within the tasks considered, the asynchronous training of QRL agents can reach performance comparable to or superior than classical agents with similar model sizes and architectures.