Vaccine hesitancy poses a significant threat to the health of individuals across all age groups, which has been exacerbated by the COVID-19 pandemic. In this cross-sectional study, an extension of the Theory of Planned Behavior (TPB) was applied to investigate psychosocial variables predicting intention to vaccinate children under 12 against COVID-19 in a sample of 420 Italian parents (Mean age = 40.4, SD = 5.9; Women = 78.1%). Hierarchical regression analysis revealed that, among the TPB variables, cognitive attitude, descriptive norms, and perceived behavioral control significantly predicted parents’ vaccination intention. Furthermore, including trust in the institutions’ ability to manage the vaccination campaign in the model significantly increased the explained variance in intention. These findings suggest that campaigns promoting childhood COVID-19 vaccination should not only emphasize the safety and effectiveness of vaccines for children but also focus on reducing barriers to vaccination. Additionally, attention should be given to enhancing the perception that this behavior is widespread among other parents, thus leveraging the power of social influence. Finally, and not less important, significant efforts should be directed toward building and reinforcing trust in the system of actors promoting and managing the COVID-19 vaccination campaign.