COVID-19 vaccination programs continue in child populations. Thus, parents’ attitude towards COVID-19 vaccination of their children is crucial for these strategies to succeed. The present study derives from the application of an online COVID-19 Vaccine Acceptance & Hesitancy Questionnaire (COV-AHQ) in which we measure parent’s hesitancy towards children’s vaccination (section 4 of the COV-AHQ) and other significant factors. A logistic regression analysis with backward stepwise method was used to quantify the associations between factors and parent’s hesitancy. According to the correlation analysis, the most representative factors predicting vaccine hesitancy/acceptance were positive attitude towards vaccination, parents believing that the COVID-19 vaccine will enhance the economic situation of the country, parents actively researching information, having the willingness to obtain the COVID-19 vaccine themselves, and the possibility of their children developing adverse effects. Our findings also showed that parents are highly interested in having their children vaccinated. Nonetheless, parents expressed high levels of concern involving their children in developing adverse effects from the vaccine. In addition, obtaining influenza immunization prompted interest in obtaining the COVID-19 vaccine, and younger-aged parents are much more concerned with having their children vaccinated. Therefore, in order to ensure successful vaccination programs, policymakers and health authorities should design strategies to gain confidence and provide security amongst the population, including giving continuous information about the benefits of vaccination and presenting the frequency of side effects to bring parents on board with vaccinating their children.