Motion planning is a crucial component of autonomous driving technology, governing vehicle movement. However, when executing automated parking in complex environments, motion planning modules face significant challenges: 1) The need for precise obstacle avoidance that satisfies vehicle kinematics in confined spaces. 2) The requirement for efficient computation under numerous non-convex collision avoidance constraints imposed by dense obstacles. To address these challenges, we propose a two-stage motion planner. This planner optimizes a coarse path generated by a hybrid A* algorithm using a model predictive control (MPC) approach. We transform the non-convex constraints imposed by obstacles into a smooth bi-convex reformulation through dual solving, and further reduce computational time by employing the Alternating Direction Method of Multipliers (ADMM) for parallel computation. We validate the planner’s performance through simulation experiments. Results demonstrate that the proposed method exhibits high-precision parking planning capabilities with enhanced computational efficiency.