The COVID-19 pandemic presents a significant challenge to the lives and well-being of families with underaged children. Although previous studies have documented COVID-related deterioration in well-being and identified protective and risk factors, the mechanisms under which the pandemic leads to worsened well-being remain unclear. In addition, from a policymaker’s perspective, it is important to differentiate between the effects of government-issued infection control measures (such as lockdown) and families’ voluntary responses when facing the coronavirus (such as self-quarantine) on well-being. Using Singapore as an example, we collected retrospective self-reports on the everyday activities, stressors, and well-being of parents and other caregivers at three timepoints: before local transmission (Pre-pandemic), after local transmission but before the “circuit breaker” (Pandemic), and during the “circuit breaker” (Lockdown). We estimated the effects of the pandemic itself and families’ voluntary responses to it by contrasting Pandemic against Pre-pandemic, and we estimated the additional effects of imposed lockdown measures by contrasting Lockdown against Pandemic. Results showed significant changes in jobs and income, childcare arrangement, family dynamics, and parents’ emotional well-being throughout the three timepoints. Both mothers and fathers reported to worry most about the health and safety of family members and self. Mothers’ time spent on housework partially mediated the effect of lockdown on their emotional well-being, and parents’ conflict with other adults in the household partially mediated the effects of both pandemic and lockdown on their emotional well-being. The effects of pandemic and lockdown were also moderated by parents’ age, education level, and fathers’ authoritarian values.