The current COVID-19 pandemic has been accompanied by the circulation of an unprecedented amount of “polluted” information, especially in the social media environment, among which are false narratives and conspiracy theories about both the pandemic and vaccination against COVID-19. The effects of such questionable information primarily concern the lack of compliance with restrictive measures and a negative attitude towards vaccination campaigns, as well as more complex social effects, such as street protests or distrust in governments and authorities in general. Even though there is a lot of scholarly attention given to these narratives in many countries, research about the profile of people who are more prone to believe or spread them is rather scarce. In this context, we investigate the role of age, compared with other socio-demographic factors (such as education and religiosity), as well as the role of the media (the frequency of news consumption, the perceived usefulness of social media, and the perceived incidence of fake information about the virus in the media) and the critical thinking disposition of people who tend to believe such misleading narratives. To address these issues, we conducted a national survey (N = 945) in April 2021 in Romania. Using a hierarchical OLS regression model, we found that people who perceive higher incidence of fake news (ß = 0.33, p < 0.001), find social media platforms more useful (ß = 0.13, p < 0.001), have lower education (ß = −0.17, p < 0.001), and have higher levels of religiosity (ß = 0.08, p < 0.05) are more prone to believe COVID-19-related misleading narratives. At the same time, the frequency of news consumption (regardless of the type of media), critical thinking disposition, and age do not play a significant role in the profile of the believer in conspiracy theories about the COVID-19 pandemic. Somewhat surprisingly, age does not play a role in predicting belief in conspiracy theories, even though there are studies that suggest that older people are more prone to believe conspiracy narratives. As far as media is concerned, the frequency of news media consumption does not significantly differ for believers and non-believers. We discuss these results within the context of the COVID-19 pandemic.