Within a few months, the scientific world achieved a great success, developing effective and safe vaccines against COVID-19. Many countries with full access to vaccines have introduced recommendations for the vaccination of not only people who are at risk of developing severe COVID-19, i.e., the elderly and chronically ill, but all members of society, including children aged 12 and above as the currently registered preparations can be used above the said age. However, the use of COVID-19 vaccines in children arouses strong emotions, with their sense being frequently questioned. The aim of the paper was to assess the attitudes of Polish parents with regard to vaccinations against COVID-19 administered to their children. The study was conducted with the use of the authors’ original questionnaire, which was distributed online. The questionnaire was completed 4732 times, and 4432 surveys were qualified for the final analysis. The vast majority of the respondents were women (77.6%), people aged 36–44 (40.3%), with higher education (86.2%) and people living in the city with more than 250,000 inhabitants (48%). The mean age of the study group was 37.5 ± 6.61 years. Out of the studied group, 28.2% of parents are healthcare professionals. The study was conducted prior to the EMA’s decision that permitted the use of BNT162b2 in children. Results: The vast majority of the respondents were mothers, who showed significantly more favorable attitudes toward the vaccinations than fathers. Forty-four percent of parents want to vaccinate their children as soon as possible, while every fourth parent does not want to vaccinate their child at all. Main concerns about the vaccines include concerns that the preparation has not been adequately tested and that it is ineffective, as well as the lack of information concerning potential complications in the future. The main sources of information on childhood vaccinations are the media, including the Internet and television. Vaccination of the pediatric population against COVID-19 raises many emotions and doubts in parents and it is also debated by experts. The decision to vaccinate should rest on child’s parents. Both the individual benefits of protection against COVID-19 and the population benefits of pandemic control must be considered. There is a need for ongoing monitoring of the safety of administering COVID-19 vaccinations in children, as well as for evaluating their effectiveness and benefits in reducing individual risk of severe course of COVID-19 and complications after this disease, and for evaluating the population benefits of vaccines in children.