In recent years, there have been significant advancements in semantic parsing tasks, thanks to the introduction of pre-trained language models. However, a substantial gap persists between English and other languages due to the scarcity of annotated data. One promising strategy to bridge this gap involves augmenting multilingual datasets using labeled English data and subsequently leveraging this augmented dataset for training semantic parsers (known as zero-shot multilingual semantic parsing). In our study, we propose a novel framework to effectively perform zero-shot multilingual semantic parsing under the support of large language models (LLMs). Given data annotated pairs (sentence, semantic representation) in English, our proposed framework automatically augments data in other languages via multilingual chain-of-thought (CoT) prompting techniques that progressively construct the semantic form in these languages. By breaking down the entire semantic representation into sub-semantic fragments, our CoT prompting technique simplifies the intricate semantic structure at each step, thereby facilitating the LLMs in generating accurate outputs more efficiently. Notably, this entire augmentation process is achieved without the need for any demonstration samples in the target languages (zero-shot learning). In our experiments, we demonstrate the effectiveness of our method by evaluating it on two well-known multilingual semantic parsing datasets: MTOP and MASSIVE.