As the trend of moving away from high-precision maps gradually emerges in the autonomous driving industry,traditional planning algorithms are gradually exposing some problems. To address the high real-time, high precision, and high trajectory quality requirements posed by the automatic parking task under real-time perceived local maps,this paper proposes an improved automatic parking planning algorithm based on the A* algorithm, and uses Model Predictive Control (MPC) as the control module for automatic parking.The algorithm enhances the planning real-time performance by optimizing heuristic functions, binary heap optimization, and bidirectional search; it calculates the passability of narrow areas by dynamically loading obstacles and introduces the vehicle's own volume during planning; it improves trajectory quality by using neighborhood expansion and Bezier curve optimization methods to meet the high trajectory quality requirements of the parking task. After obtaining the output results of the planning algorithm, a loss function is designed according to the characteristics of the automatic parking task under local maps, and the MPC algorithm is used to output control commands to drive the car along the planned trajectory. This paper uses the perception results of real driving environments converted into maps as planning inputs to conduct simulation tests and ablation experiments on the algorithm. Experimental results show that the improved algorithm proposed in this paper can effectively meet the special requirements of automatic parking under local maps and complete the automatic parking planning and control tasks.