We propose $\text{GMP}^{3}$, a multiphase global path planning framework that generates dynamically feasible three-dimensional trajectories for unmanned aerial vehicles (UAVs) operating in cluttered environments. The framework extends traditional path planning from Euclidean position spaces to the Lie group $\mathrm{SE}(3)$, allowing joint learning of translational motion and rotational dynamics. A modified Bellman-based operator is introduced to support reinforcement learning (RL) policy updates while leveraging prior trajectory information for improved convergence. $\text{GMP}^{3}$ is designed as a distributed framework in which agents influence each other and share policy information along the trajectory: each agent refines its assigned segment and shares with its neighbors via a consensus-based scheme, enabling cooperative policy updates and convergence toward a path shaped globally even under kinematic constraints. We also propose DroneManager, a modular ground control software that interfaces the planner with real UAV platforms via the MAVLink protocol, supporting real-time deployment and feedback. Simulation studies and indoor flight experiments validate the effectiveness of the proposed method in constrained 3D environments, demonstrating reliable obstacle avoidance and smooth, feasible trajectories across both position and orientation. The open-source implementation is available at https://github.com/Domattee/DroneManager