Real-time path planning in constrained environments remains a fundamental challenge for autonomous systems. Traditional classical planners, while effective under perfect perception assumptions, are often sensitive to real-world perception constraints and rely on online search procedures that incur high computational costs. In complex surroundings, this renders real-time deployment prohibitive. To overcome these limitations, we introduce a Deep Reinforcement Learning (DRL) framework for real-time path planning in parking scenarios. In particular, we focus on challenging scenes with tight spaces that require a high number of reversal maneuvers and adjustments. Unlike classical planners, our solution does not require ideal and structured perception, and in principle, could avoid the need for additional modules such as localization and tracking, resulting in a simpler and more practical implementation. Also, at test time, the policy generates actions through a single forward pass at each step, which is lightweight enough for real-time deployment. The task is formulated as a sequential decision-making problem grounded in a bicycle model dynamics, enabling the agent to directly learn navigation policies that respect vehicle kinematics and environmental constraints in the closed-loop setting. A new benchmark is developed to support both training and evaluation, capturing diverse and challenging scenarios. Our approach achieves state-of-the-art success rates and efficiency, surpassing classical planner baselines by +96% in success rate and +52% in efficiency. Furthermore, we release our benchmark as an open-source resource for the community to foster future research in autonomous systems. The benchmark and accompanying tools are available at https://github.com/dqm5rtfg9b-collab/Constrained_Parking_Scenarios.