Automated parking is a self-driving feature that has been in cars for several years. Parking assistants in currently sold cars fail to park in more complex real-world scenarios and require the driver to move the car to an expected starting position before the assistant is activated. We overcome these limitations by proposing a planning algorithm consisting of two stages: (1) a geometric planner for maneuvering inside the parking slot and (2) a Rapidly-exploring Random Trees (RRT)-based planner that finds a collision-free path from the initial position to the slot entry. Evaluation of computational experiments demonstrates that improvements over commonly used RRT extensions reduce the parking path cost by 21 % and reduce the computation time by 79.5 %. The suitability of the algorithm for real-world parking scenarios was verified in physical experiments with Porsche Cayenne.