Introduction Understanding parents’ and children’s mental health issues would help design population-specific intervention programs. The present study explored parents’ perceived stress and child emotions and behavior during the COVID-19 lockdown among Sri Lankan families. Methods A cross-sectional survey was conducted among Sri Lankan parents of children aged 11 to 17 years. Validated instruments (Perceived Stress Scale-PSS and Strengths and Difficulties Questionnaire-SDQ) evaluated parental stress, child emotions, and hyperactivity/inattention. Multiple linear regression assessed the predictors of mental health issues, including the interaction between age and gender. Results Three hundred fifty-five parents responded to the survey (mothers:76%). One-third of parents experienced difficulties with their children during the pandemic. Emotions and hyperactivity-inattention problems measured via the SDQ scale were high among 38% of children, while the perceived stress was high in 79.2% of parents. Overall, child emotions and hyperactivity-inattention increased with decreasing age, increasing parent stress, having middle-income compared to high-income, and having a family member/close relative tested positive for COVID-19. Hyperactivity-inattention (29.3%) was more than the emotional problems (22%) among children. The emotional problems were reported more with increasing parent stress, while child hyperactivity-inattention alone was reported more with decreasing age, middle-income compared to high-income families, and increasing parent stress. Also, the interaction effect of age and gender indicated that higher age was related to greater parent-reported hyperactivity-inattention problems in males. Conclusions The findings highlight how the COVID-19 crisis and social isolation have contributed to increased parental stress and child emotional and hyperactivity-inattention problems. In addition to cautioning the healthcare workers, socio-culturally appropriate preventive and supportive mental health programs may help deal with further waves of COVID-19 or any other adverse circumstances.