The aim of the present study was to assess the psychological impact of hospitalization during the COVID-19 pandemic on parents and their offspring. We performed a nationwide cross-sectional study in Greece based on an Internet questionnaire survey. A convenience sample of parents whose offspring had been hospitalized due to COVID-19 (including multisystem inflammatory syndrome in children, MIS-C), diagnosed with COVID-19 but not hospitalized, and hospitalized for another reason during the pandemic were enrolled. Parental stress was assessed using the Perceived Stress Scale (PSS) and the Revised Impact of Event Scale (IES-R) tools, and childhood mental wellbeing with the Children’s Revised Impact of Event 13 (CRIES-13) scale. Out of 214 received responses, stress levels were significantly higher in parents whose children had been admitted for COVID-19 or MIS-C versus those not admitted or admitted for other reasons (p < 0.001, for PSS/IES-R). Parental and childhood stress levels were correlated. In the multivariable linear regression analysis, children’s hospitalization because of COVID-19 or MIS-C, younger parental age, the existence of comorbidities, and another family member’s hospitalization because of COVID-19 were independent factors for higher stress. In light of the above, stricter hospital admission criteria for COVID-19 could be implemented, and psychological support for eventually admitted families may be beneficial.