Throughout the COVID-19 period, families were forced to stay indoors, adapting to online schooling, remote work, and virtual social engagements, inevitably altering the dynamics within households. There was a notable increase in mental health challenges in terms of anxiety and depression in children and adolescents. This study intended to explore the psychosocial effects of the COVID-19 pandemic on Italian families by adopting self- and proxy-report questionnaires on anxiety, anger, and health-related quality of life. The results showed that approximately 20% obtained a clinical anxiety score and only 10% obtained a clinical anger score. There was a difference in the perception of the quality of life reported by the child and that perceived by the parent. A stepwise regression model showed that total anxiety scores were predicted by sex, quality of life scores from the parents’ self-report version, and the total anger score. Another stepwise regression model identified physiological and social anxiety as the best predictors that impact quality of life. Parental well-being actively influences the well-being of children, so it is fundamental to implement preventive programs and promote child well-being by providing parents the most adequate support possible.