This paper presents a quasi-linear parameter-varying (quasi-LPV) control framework for the cooperative transportation of a cable-suspended payload using multiple quadrotors. The main objective is to exploit systematic linear-system control techniques to reduce the complexity of the controller design while retaining the relevant nonlinear dependencies of the multi-vehicle system. The proposed architecture is organized into two interconnected levels: a payload-level force allocation stage and a vehicle-level quasi-LPV control stage. At the payload level, the required resultant force is distributed among the suspension cables through a regularized least-squares formulation. At the vehicle level, the nonlinear quadrotor dynamics are represented by a polytopic quasi-LPV model using measurable scheduling variables. An integral state-feedback controller is synthesized through linear matrix inequalities (LMIs), incorporating an $H_{\infty }$ criterion to attenuate the effect of cable-induced interaction forces on the tracking performance. Numerical simulations are performed for formations composed of three, five, and ten quadrotors following circular, figure-eight, and three-dimensional reference trajectories, respectively. The results show accurate payload and vehicle trajectory tracking, bounded cable-force allocation errors, and consistent performance as the number of cooperating vehicles increases. These results demonstrate that the quasi-LPV formulation enables linear control-design tools to be effectively applied to a nonlinear and strongly coupled cooperative aerial transportation system.