This paper presents a fast optimization based algorithm for car parking. The challenge arises from the non-holonomic characteristics of the car and the close distance to the obstacles. The presented approach utilizes the Minkowski sum to account for obstacle avoidance. The geometric path planning problem is decoupled from the kinematic problem and discretized with respect to the path parameter by means of a Runge-Kutta discretization. For the discrete path segments, an optimization problem is formulated to calculate the path independent of the parking scenario. This static optimization problem can be solved numerically in a very efficient way. The performance of the algorithm is evaluated in several simulation scenarios.