Different COVID-19 vaccines have been approved for underage children, so parents and caregivers currently face the decision of whether to vaccinate their children against COVID-19 or not. Due to the rather moderate vaccine acceptance among parents across different countries, the objective of the present study was to investigate the relationship between different psychological, demographic, and behavioral factors related to the acceptance of the COVID-19 vaccine for underage children among parents. In particular, vaccination attitudes, whether parents have been vaccinated against COVID-19 themselves, COVID-19 fear, attitude towards COVID-19 policy measures, governmental trust, subjective level of information, perceived risk of disease progression, and perceived risk of vaccine side effects were the variables of interest. The study adopted a cross-sectional study design, and the sample consisted of 2405 participants. A network analysis was conducted to investigate the associations and interconnection among these variables. The results showed that, in particular, compliance, confidence in the safety of vaccines, whether parents have been vaccinated against COVID-19 themselves, trust in the governmental system, fear of COVID-19, and the parents’ age were directly related to the acceptance of the COVID-19 vaccine for children. To increase compliance and confidence in the vaccines’ safety among parents, promotion campaigns should provide more information concerning the vaccines’ safety, particularly for younger parents who are not vaccinated against COVID-19 themselves.