In this paper, we propose an algorithm for optimal generation of nonholonomic paths for planning parking maneuvers with a kinematic car model. We demonstrate the use of Successive Convexification algorithms (SCvx), which guarantee path feasibility and constraint satisfaction, for parking scenarios. In addition, we formulate obstacle avoidance with state-triggered constraints which enables the use of logical constraints in a continuous formulation of optimization problems. This paper contributes to the optimal nonholonomic path planning literature by demonstrating the use of SCvx and state-triggered constraints which allows the formulation of the parking problem as a single optimisation problem. The resulting algorithm can be used to plan constrained paths with cusp points in narrow parking environments.