(1) Background: this study is based on a model of how changes in protective factors may affect the emotional health of mothers and fathers and thus influence the development of the baby. Our research goal is to determine whether variations in perceived social support moderate levels of stress and depression during pregnancy and/or the effect of parents’ emotional health on the baby’s anthropometric parameters. (2) Methods: to achieve these aims, a longitudinal study was made of 132 couples and babies, who were evaluated at weeks 12 and 32 of gestation and at birth. Separate analyses were performed for the mothers and fathers, focused on the role of social support in moderating their levels of depression and stress during pregnancy, and the consequent impact on the baby. (3) Results: the results obtained show the moderating effects of changes in social support on maternal and paternal stress and depression. Reduced social support during pregnancy is associated with higher levels of stress and depression in both parents and with a high cephalisation index in their babies. (4) Conclusions: special attention should be paid to social support, which can have a strong impact on the evolution of emotional health during pregnancy and concomitantly on the development of the baby.