Purpose: Once vaccination against COVID-19 was also possible for children over 12 years of age, parents/legal guardians had to give their consent for their vaccination. It is a crucial moment, given the large number of infected people in Romania and the fact that these children are a source of transmission of the virus in the community. The refusal or hesitation of the parents/legal guardians, regarding the agreement for the vaccination of the children, determined us to focus on this subject, wishing, based on the questioning of as many parents as possible, to extract the reasons underlying these decisions. Methods: This study is designed to observe the attitudes of parents/legal guardians regarding the refusal, hesitation, or acceptance of vaccination of children. The persons targeted to answer the questionnaire had to meet three conditions: to be at least 21 years old, to have a stable residence in Romania, and to be parents/legal guardians of at least one child under 18 years of age. The questionnaire was applied online to a number of 581 parents/legal guardians, being structured to obtain socio-demographic data and other categories of data that allow us to analyze their views on vaccinating children. Results: Sociological data resulting from the application of the questionnaire on 581 parents/legal guardians show that 183 (31.5%) adults and 140 (24.1%) children got infected with COVID-19. The total number of respondents shows that only 411 (70.7%) adults and 185 (31.8%) children are vaccinated. Conclusions: From the analysis of the data obtained through the questionnaire, following the application of the Kendall and Spearman statistical analysis tests, it is found that there is a strong link between participants’ trust/distrust in “fake news” information and their decision to vaccinate their children.