Proper Orthogonal Decomposition (POD) is a widely used technique for reduced-order modeling, capable of extracting key physical features to enable rapid analysis. While distributed memory parallel computing based on domain decomposition is effective for applying POD to large-scale problems, Local POD [1] has been proposed to further enhance computational efficiency by constructing bases for each partitioned subdomain. In this study, we propose a distributed memory parallel computation technique for Local POD to fundamentally address memory usage and computational bottlenecks. Our approach is based on generalized hierarchical graph decomposition, utilizing a multi-level graph partitioning strategy that decouples POD computation subdomains from parallel computation subdomains. The framework consists of three layers: (i) mesh nodes, (ii) POD computation subdomains for local basis generation, and (iii) parallel computation subdomains assigned to parallel processes. By separating the domain decomposition for POD basis generation from that for parallel processing, the framework achieves robust parallel efficiency across diverse problem settings. Furthermore, the integration of adaptive basis selection, hyper-reduction [2], and load-balancing techniques enhances the method's flexibility and scalability. The framework is verified through large-scale electromagnetic analysis governed by Maxwell's equations. Numerical experiments demonstrate high parallel efficiency throughout the online phase, which is further improved by the integrated load-balancing techniques.