Vaccinating children against COVID-19 is an essential public health strategy in order to reach herd immunity and prevent illness among children and adults. Parents are facing tremendous stress in relation to the COVID-19 pandemic and the effectiveness of the COVID-19 vaccination program for children. In this study, we aimed to investigate parents’ perceptions and acceptance of the COVID-19 vaccine for their children in Saudi Arabia. A well-designed, pre-validated, Google questionnaire was distributed to parents through social media websites. The selection of the participants was based on the simple random sample technique. The study sample size was 1304 participants, with 342 males (26.2%), and 962 females (73.8%). The personal information, perception about COVID-19, and their children’s vaccination status were obtained. Among the participants, 602 (46.1%) were willing to get the COVID-19 vaccination for their children, whereas 382 (29.3%) were hesitant to inoculate their children for COVID-19 and 320 (24.4%) were unsure. Age (p = 0.004), gender (p = 0.001), occupation (0.004), income (p = 0.030), and vaccination status (p = 0.001) had an influence on the parents’ acceptance of COVID-19 vaccination of their children. On the other hand, education level, number of children, and having been previously infected with COVID-19 had no statistically significant effect on the parent acceptance. The correlation of parents’ knowledge about COVID-19 and their agreement to the vaccination of their children was statistically significant, along with gender (males were more knowledgeable, with p < 0.001), occupation, income (higher income showed a statistical difference, with p < 0.001), and vaccination status (p < 0.001). There was a decrease in parents’ acceptance toward the COVID-19 children vaccine in Saudi Arabia, which requires more attention and focus from health providers to eliminate fear and anxiety among the parents through additional educational programs and events to decrease the resistance toward the vaccination of children. More emphasis is required to increase the awareness of parents and convey the importance of the vaccine for children. In addition, more studies are needed to ensure the vaccine’s safety.