The international literature has shown that maternal and paternal postnatal depression (PND) is one of the most common mental illnesses in the perinatal period, with significant consequences for parent–infant relationships and infant development. The COVID-19 pandemic has increased the rates of prevalence of PND, exacerbating the mental health risk for new families. This systematic review aims to examine the effect of maternal and paternal PND on parent–infant relationships and children’s development in the first 36 months after childbirth during the COVID-19 outbreak. Eligible studies were identified using the following databases: Medline, CINAHL, SCOPUS, PsycINFO, PsycARTICLES, ScienceDirect, and Web of Science. Of the 1252 studies considered, 10 studies met the inclusion criteria. Results showed that maternal PND significantly affected the quality of the early mother–infant relationship and the infant’s motor, self-regulation, and socio-emotional development. In addition, the detrimental impact of maternal PND on the quality of early mother–infant relationships seems to become stronger as COVID-19 concerns increase. No studies included fathers. These findings strengthened the importance of planning targeted prevention and treatment strategies to prevent PND and its short- and long-term consequences, especially in the case of stressful and traumatic events. They also suggested the urgent need for further exploration of fathers.