Autonomous drone racing has gained attention for its potential to push the boundaries of drone navigation technologies. While much of the existing research focuses on racing in obstacle-free environments, few studies have addressed the complexities of obstacle-aware racing, and approaches presented in these studies often suffer from overfitting, with learned policies generalizing poorly to new environments. This work addresses the challenge of developing a generalizable obstacle-aware drone racing policy using deep reinforcement learning. We propose applying domain randomization on racing tracks and obstacle configurations before every rollout, combined with parallel experience collection in randomized environments to achieve the goal. The proposed randomization strategy is shown to be effective through simulated experiments where drones reach speeds of up to 70 km/h, racing in unseen cluttered environments. This study serves as a stepping stone toward learning robust policies for obstacle-aware drone racing and general-purpose drone navigation in cluttered environments. Code is available at https://github.com/ErcBunny/IsaacGymEnvs.