CO-FAST presents a novel ground-braking framework for cooperative aerial manipulation based on swarms of unmanned aerial vehicles connected to a shared payload through motorized, extensible cables. The system is modeled as a floating-base parallel kinematic structure, enabling full six-degree-of-freedom payload manipulation while significantly simplifying swarm formation control. By employing independently actuated winch-driven cables, CO-FAST decouples payload motion from the geometric configuration of the aerial swarm, overcoming key limitations of traditional fixed-cable approaches that rely solely on robot positioning. This decoupling can be exploited within a hierarchical optimization-based control architecture that balances stability, energy efficiency, and manipulation capability. Unlike quasi-static methods, the proposed framework can explicitly account for dynamic couplings and external disturbances, enabling responsive and robust control. The capabilities of CO-FAST are demonstrated by two high-impact application scenarios: cooperative aerial wildfire suppression, leveraging game-theoretic resource allocation strategies, and precise aerial transportation and assembly of heavy or asymmetric loads in construction environments. In these contexts, the proposed approach shows the potential to outperform conventional ground-based solutions in terms of accessibility, deployment time, and operational cost. By combining the maneuverability of aerial platforms with the precision of cable-driven parallel robotics, CO-FAST advances the state of the art in distributed aerial manipulation and contributes to the development of sustainable, flexible solutions for emergency response and infrastructure operations.