Distributed optimization offers a promising paradigm for trajectory planning in Unmanned Aerial Vehicle (UAV) swarms, yet its deployment in communication-constrained environments remains challenging due to unreliable links and limited data exchange. This paper addresses this issue via a two-tier architecture explicitly designed for operation under communication constraints. We develop a Communication-Aware Asynchronous Distributed Trajectory Optimization (CA-ADTO) framework that integrates Parameterized Differential Dynamic Programming (PDDP) for local trajectory optimization of individual UAVs with an asynchronous Alternating Direction Method of Multipliers (async-ADMM) for swarm-level coordination. The proposed architecture enables fully distributed optimization while substantially reducing communication overhead, making it suitable for real-world scenarios in which reliable connectivity cannot be guaranteed. The method is particularly effective in handling nonlinear dynamics and spatio-temporal coupling under communication constraints.