\textit{Model Parsing} defines the task of predicting hyperparameters of the generative model (GM), given a GM-generated image as the input. Since a diverse set of hyperparameters is jointly employed by the generative model, and dependencies often exist among them, it is crucial to learn these hyperparameter dependencies for improving the model parsing performance. To explore such important dependencies, we propose a novel model parsing method called Learnable Graph Pooling Network (LGPN), in which we formulate model parsing as a graph node classification problem, using graph nodes and edges to represent hyperparameters and their dependencies, respectively. Furthermore, LGPN incorporates a learnable pooling-unpooling mechanism tailored to model parsing, which adaptively learns hyperparameter dependencies of GMs used to generate the input image. Also, we introduce a Generation Trace Capturing Network (GTC) that can efficiently identify generation traces of input images, enhancing the understanding of generated images' provenances.Empirically, we achieve state-of-the-art performance in model parsing and its extended applications, showing the superiority of the proposed LGPN.