As the global vaccination mass campaign against COVID-19 extended to children aged 5 to 11 years, some parents remained hesitant about their children being administered the vaccine despite data supporting its safety. Parent vaccine hesitancy (PVH) may have predisposed certain groups of children, particularly those with autism spectrum disorder (ASD), to COVID-19 when other neurotypical children would have been vaccinated. We investigated the current PVH in 243 parents of children with ASD and 245 controls using the Parent Attitudes about Childhood Vaccines (PACV) scale. The study was conducted in Qatar from May to October 2022. Overall, 15.0% [95% CI 11.7%; 18.3%] of parents were vaccine-hesitant, with no difference (p = 0.054) between groups (ASD children [18.2%] vs. controls [11.7%]). The only sociodemographic factor associated with higher vaccine hesitancy was being a mother (as compared to being a father). The COVID-19 vaccine receipt rate at the time of the study did not differ between ASD (24.3%) and non-ASD groups (27.8%). Around two-thirds of parents of children with ASD refused or were unsure about vaccinating their children against COVID-19. We found that the intent to vaccinate against COVID-19 was higher in parents who were married and in those with a lower PACV total score. Continued public health efforts are needed to address vaccine hesitancy among parents.