Unmanned Aerial Vehicles (UAVs) have great potential to achieve a variety of tasks remotely such as aerial grasping, transporting and manipulating objects. Architectures with multiple UAVs have further enhanced the payload capacity and manipulability of these robots, for instance a Flying Parallel Robot (FPR) where a moving platform is cooperatively supported by multiple quadrotors with passive rigid links. In this paper, we address the vision-based state estimation and decentralized control applied to the multi-UAV parallel robot, taking the FPR as an example. An ArUco marker system is applied to estimate the relative pose of each UAV with respect to the common platform frame, along with the Extended Kalman Filter to reconstruct the robot state without the dependence on any external localization system. The interaction controller is then deployed in a decentralized manner, which is potentially more robust to communication delays or interruptions. The proposed methodology has been validated by real-time experiments demonstrating the teleoperation of the FPR interacting with the environment.