The focus of coronavirus disease 2019 (COVID-19) vaccination campaigns has been the adult population, particularly the elderly and those with chronic diseases. However, COVID-19 can also affect children and adolescents. Furthermore, targeting this population can accelerate the attainment of herd immunity. The aim of the current study was to evaluate parental intentions to vaccinate their children and the variables associated with them. An online questionnaire was circulated via generic Iraqi Facebook groups to explore parental intentions regarding the vaccination of their children. Multinomial regression analysis was conducted to evaluate variables associated with parental vaccination acceptance. A total of 491 participants completed the study questionnaire. Only 38.3% of the participants were willing to vaccinate their children against COVID-19, while the rest either refused to vaccinate their children (35.6%) or were unsure whether they would (26.1%). Participants’ perceptions about the effectiveness (OR = 0.726, 95% CI = 0.541–0.975, p = 0.033) and safety (OR = 0.435, 95% CI = 0.330–0.574, p < 0.0001) of COVID-19 vaccines were significantly associated with parental acceptance of having children vaccinated. Participants who had received or who were planning to receive the COVID-19 vaccine were significantly less likely to reject vaccinating their children (OR = 0.156, 95% CI = 0.063–0.387, p < 0.0001). There is high refusal/hesitancy among Iraqi parents to vaccinate their children, which is associated with concerns related to the safety and efficacy of COVID-19 vaccines