Nowadays, the autonomous parking and park assist systems are getting more attention since they provide comfort and relief to the drivers and prevent parking-lot accidents. In this paper, we overview several main geometric path planning methods for autonomous parking of non-holonomic vehicles and we analyze their performances with respect to the final lateral distance to the parking spot. To the best of our knowledge, this kind analysis has never been done before for autonomous parallel parking algorithms. Moreover, we provide a new geometric method to reduce the final lateral distance between the vehicle and the parking spot. While the reference methods are insufficient for the final lateral distance, the proposed method introduces a realizable solution for long vehicles and enhances the performance for short vehicles. Simulations which are performed with various types of vehicles provide up to 48% better performance and show the efficiency of the new approach clearly.