Background: The way postpartum parents’ COVID-19-related concerns are associated with the family environment, support resources and depressive symptoms areunder-investigated. Methods: Two hundred and forty-three new parents (132 mothers, 111 fathers) completed self-report questionnaires within an 8-week period after birth. Parental concerns for COVID-19-related life changes were assessed with the COVID-19 Questionnaire, perceived social support with the Multidimensional Scale of Perceived Social Support, perceived family functioning with the Family Adaptability and Cohesion Evaluation Scales IV Package, dyadic coping behaviors with the Dyadic Coping Inventory and maternal/paternal postnatal depression with the Edinburgh Postnatal Depression Scale. Results: (a) Higher levels of COVID-19-related concerns about daily life were associated with lower levels of family communication, satisfaction and increased depressive symptomatology in both parents, and with lower levels of family functioning in mothers; (b) Maternal health care COVID-19-related concerns were linked with lower levels of family communication, lower perceived social support and with an increase in maternal depressive symptoms; and (c) COVID-19-related concerns about neonate hospitalization were associated with increased maternal depressive symptoms. Conclusion: Τhese findings suggest that COVID-19-related concerns had a common negative effect on both postpartum mothers’ and fathers’ mental health and on certain aspects of family functioning.