The integration of shared mobility and autonomous vehicles (AVs) could potentially change the way parents decide to transport their children to and from school. A better understanding of the factors influencing parents’ intentions to use shared autonomous vehicles (SAVs) for school transportation is necessary to enhance their children’s mobility. Unlike prior research, this paper significantly contributes to the literature by exploring the impacts of socioeconomic, travel-related, and psychological factors and their interactions. Using Google Forms for an online survey, the authors collected 1435 valid responses from parents in Kerman city schools in Iran. The estimation results of the generalized ordered logit model indicate the significant impact of parents’ socioeconomic status (occupation, education, income), travel behavior (accident experience, crash severity, travel cost), and attitude (innovativeness, perceived usefulness, environmental concern, pro-driving, safety), and their children-related factors (gender, the most frequently used travel mode, the possibility of tracking the child). The findings show that an increase in parents’ education, perceived usefulness, and environmental concern increase the likelihood of their intentions to use SAVs. Finally, based on the findings, several implications are suggested to increase parents’ intentions to use SAVs for transporting their children and to make SAVs a safe, affordable, and sustainable transport solution.