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Text-based person search (TBPS) aims to retrieve the images of the target person from a large image gallery based on a given natural language description. Existing methods are dominated by training models with parallel image-text pairs, which are very costly to collect. In this paper, we make the first attempt to explore TBPS without parallel image-text data ($\mu$-TBPS), in which only non-parallel images and texts, or even image-only data, can be adopted. Towards this end, we propose a two-stage framework, generation-then-retrieval (GTR), to first generate the corresponding pseudo text for each image and then perform the retrieval in a supervised manner. In the generation stage, we propose a fine-grained image captioning strategy to obtain an enriched description of the person image, which firstly utilizes a set of instruction prompts to activate the off-the-shelf pretrained vision-language model to capture and generate fine-grained person attributes, and then converts the extracted attributes into a textual description via the finetuned large language model or the hand-crafted template. In the retrieval stage, considering the noise interference of the generated texts for training model, we develop a confidence score-based training scheme by enabling more reliable texts to contribute more during the training. Experimental results on multiple TBPS benchmarks (i.e., CUHK-PEDES, ICFG-PEDES and RSTPReid) show that the proposed GTR can achieve a promising performance without relying on parallel image-text data.