Safely handling occlusions is a fundamental challenge for autonomous mobile robots operating in dynamic environments. This issue is especially prominent in autonomous valet parking (AVP), where traffic rules are lax, occlusions are frequent and cluttered, and overly conservative behavior can leave vehicles stuck. However, existing methods either lack formal safety guarantees, assume agents follow road structures, or introduce conservatism, leaving occlusion-aware planning for AVP an open challenge. In this paper, we propose APRO (AH-Polyhedron Reachability for Occlusions), an exact and efficient occlusion-aware planning framework based on game-theoretic active perception and AH-polyhedron reachability analysis with AVP as our canonical use case. Our key insight is to reformulate set-based safety conditions in prior work as unions of AH-polyhedrons, enabling exact safety verification through linear programming (LP) without any additional conservatism in set computations or assumptions on road topology. We further show how the resulting safety conditions can be integrated into optimization-based planners or a bisection search scheme for real-time applications. We validate our method in simulation and hardware experiments, including data replay on a real-world parking lot dataset. Experimental results demonstrate that our method consistently achieved a 100% safety rate across all evaluated scenarios while maintaining real-time performance, resulting in safer and more optimal decisions than existing methods with formal safety guarantees.