COVID-19 vaccination has been extended to include children aged 5–11 years. This cross-sectional survey evaluated parental COVID-19 vaccine willingness and hesitancy, and associated factors, for their children aged 5–11 years with chronic conditions. A telephone survey was conducted from 14 December 2021 to 4 January 2022. The questionnaire assessed participants’ socio-demographic and health-related information, attitudes towards COVID-19 infection, hesitancy, by using the PACV-5 (Parent Attitudes About Childhood Vaccines Survey Tool), and sources of information. A total of 430 answers were collected anonymously. Respondents with no cohabitant who had been infected by SARS-CoV-2 and having been vaccinated against COVID-19 had a higher concern about the severity of COVID-19. The parents’ perceived risk that the child could be infected by SARS-CoV-2 was higher in those more concerned about the severity of COVID-19, with an older child, and who had at least one cohabitant positive for COVID-19. Only 38.8% parents were willing to vaccinate their children against COVID-19. Parents who did not need additional information, those with higher education, those who have been vaccinated against COVID-19, those whose child was older, who had received information on this vaccination from physicians, with higher self-reported concern about the severity of COVID-19, and who had a higher perceived risk that their child could be infected by SARS-CoV-2, expressed a greater willingness to vaccinate their child. Overall, 26.3% were high-hesitant, with a PACV-5 score ≥ 7. Respondents who did not get the COVID-19 vaccine, were less educated, with a lower concern about severity of COVID-19, and with a lower perceived risk that their child could be infected by SARS-CoV-2, were more likely to be high-hesitant. New policies and educational programs regarding COVID-19 vaccination for children with chronic conditions are needed to reduce hesitancy and increase vaccination uptake.