Background: When COVID-19 was declared as a pandemic, many countries imposed severe lockdowns that changed families' routines and negatively impacted on parents' and children's mental health. Several studies on families with children with autism spectrum disorder (ASD) revealed that lockdown increased the difficulties faced by individuals with ASD, as well as parental distress. No studies have analyzed the interplay between parental distress, children's emotional responses, and adaptive behaviors in children with ASD considering the period of the mandatory lockdown. Furthermore, we compared families with children on the spectrum and families with typically developing (TD) children in terms of their distress, children's emotional responses, and behavioral adaptation. Methods: In this study, 120 parents of children aged 5–10 years (53 with ASD) participated. Results: In the four tested models, children's positive and negative emotional responses mediated the impact of parental distress on children's playing activities. In the ASD group, parents reported that their children expressed more positive emotions, but fewer playing activities, than TD children. Families with children on the spectrum reported greater behavioral problems during the lockdown and more parental distress. Conclusions: Our findings inform the interventions designed for parents to reduce distress and to develop coping strategies to better manage the caregiver–child relationship.