(1) Background: Vaccine hesitancy remains a major public health concern. The reasons behind this attitude are complex and warrant careful consideration, especially in the context of the COVID-19 era. The purpose of this study was to estimate vaccine hesitancy towards the established childhood immunization programmes in a non-random sample of Greek parents and explore possible links with important drivers of this phenomenon. (2) Methods: An online self-administered questionnaire was used from October 2020 to April 2021 to collect socio-demographic, lifestyle, and health status data and evaluate knowledge, views, and attitudes of the Greek population on COVID-19 pandemic-related issues. Parents were further asked to complete the Parent Attitudes about Childhood Vaccines (PACV) questionnaire. (3) Results: A total of 1095 parents participated in the study with a mean age of 50 years (SD 9.5 years). The hesitancy against the established childhood vaccinations was estimated at 8.9% (95% CI, 7.3–10.8%). Married status and higher education and income were negatively correlated with hesitancy, whereas positive correlations were found for stress and depressive symptoms and current smoking. Variables related to proper awareness, sound knowledge, and trust toward authorities regarding the COVID-19 pandemic were strongly associated with being less hesitant against the established childhood vaccination programmes. (4) Conclusion: The estimated parental hesitancy against the established childhood vaccination programmes is worrisome. Variables related to good awareness and knowledge of the COVID-19 pandemic were strongly associated with being less hesitant against childhood vaccinations. Since controversy surrounding COVID-19 vaccinations may decrease parents’ confidence in routine childhood vaccinations, appreciating the complex reasons behind vaccine hesitancy may inform public health policies to overcome barriers and increase vaccine acceptance.