Deep learning (DL) frameworks serve as the backbone for a wide range of artificial intelligence applications. However, bugs within DL frameworks can cascade into critical issues in higher-level applications, jeopardizing reliability and security. While numerous techniques have been proposed to detect bugs in DL frameworks, research exploring common API patterns across frameworks and the potential risks they entail remains limited. Notably, many DL frameworks expose similar APIs with overlapping input parameters and functionalities, rendering them vulnerable to shared bugs, where a flaw in one API may extend to analogous APIs in other frameworks. To address this challenge, we propose MirrorFuzz, an automated API fuzzing solution to discover shared bugs in DL frameworks. MirrorFuzz operates in three stages: First, MirrorFuzz collects historical bug data for each API within a DL framework to identify potentially buggy APIs. Second, it matches each buggy API in a specific framework with similar APIs within and across other DL frameworks. Third, it employs large language models (LLMs) to synthesize code for the API under test, leveraging the historical bug data of similar APIs to trigger analogous bugs across APIs. We implement MirrorFuzz and evaluate it on four popular DL frameworks (TensorFlow, PyTorch, OneFlow, and Jittor). Extensive evaluation demonstrates that MirrorFuzz improves code coverage by 39.92% and 98.20% compared to state-of-the-art methods on TensorFlow and PyTorch, respectively. Moreover, MirrorFuzz discovers 315 bugs, 262 of which are newly found, and 80 bugs are fixed, with 52 of these bugs assigned CNVD IDs.