Children with Autism Spectrum Disorder (ASD) often seek comfort from devices (e.g., smartphones) to deal with social overstimulation. However, such reliance exposes them to inappropriate digital content and increases susceptibility to mimicry and social vulnerability. Thus, parents having children with ASD encounter unique challenges in regulating their device usage, which are little addressed in the existing literature on parental mediation. As we begin to address this gap, we designed low-fidelity prototypes centered around open communication and self-regulation, which we refined based on the feedback from six ASD experts in two focus groups. We evaluated updated designs (presented in the form of storyboards) through semi-structured interviews with 25 parents whose children with ASD (aged below 14) are active Internet users. Our study joins the body of work on parental mediation; our findings provide insights into inclusive parental control tools for children with ASD, and offer guidelines for future research in these directions.