Autonomous Micro Aerial Vehicles (MAVs), particularly quadrotors, have shown significant potential in assisting humans with tasks such as construction and package delivery. These applications benefit greatly from the use of cables for manipulation mechanisms due to their lightweight, low-cost, and simple design. However, designing effective control and planning strategies for cable-suspended systems presents several challenges, including indirect load actuation, nonlinear configuration space, and highly coupled system dynamics. In this paper, we introduce a novel event-triggered distributed Nonlinear Model Predictive Control (NMPC) method specifically designed for cooperative transportation involving multiple quadrotors manipulating a cable-suspended payload. This approach addresses key challenges such as payload manipulation, inter-robot separation, obstacle avoidance, and trajectory tracking, all while optimizing the use of computational and communication resources. By integrating an event-triggered mechanism, our NMPC method reduces unnecessary computations and communication, enhancing energy efficiency and extending the operational range of MAVs. The proposed method employs a lightweight state vector parametrization that focuses on payload states in all six degrees of freedom, enabling efficient planning of trajectories on the SE(3) manifold. This not only reduces planning complexity but also ensures real-time computational feasibility. Our approach is validated through extensive simulation, demonstrating its efficacy in dynamic and resource-constrained environments.