Neurodevelopmental disorders (NDD) are a group of conditions affecting children’s neurodevelopment with consequences on personal, social, and educational functioning. Social robots have been used in the rehabilitation of children with NDD with encouraging results on learning outcomes. This study aims at understanding how a social robot should act to support caregivers during the rehabilitation of children with NDD. Through a Design-Based-Research approach, we investigate this question by considering the point of view of the most concerned and expert people, i.e., children with NDD and their caregivers. We present here the collaborative and iterative design of R2C3, a social robot used to support caregivers and children during rehabilitation sessions in a learning-by-teaching scenario. 27 caregivers and 6 children participated in the iterative design and/or the evaluation of R2C3, that resulted in the development of a Wizard-of-Oz interface and a library containing 120 robot behaviors. We then studied how caregivers used such behaviors during the rehabilitation sessions. We found they mainly used the robot to provide positive reinforcements to children, to elicit their reflection and knowledge toward shared handwriting activities, and to support children’s error acceptance. However, the utilization of Positive Reinforcement by caregivers tends to decrease significantly as the sessions progress.