(1) Introduction: The aim of our research was to explore emotional/behavioral changes in adolescents with neuropsychiatric conditions during the COVID-19 pandemic, and parental stress levels through a standardized assessment, comparing the data collected before and during the first months of lockdown. Moreover, an additional goal was to detect a possible relationship between emotional/behavioural symptoms of adolescents and the stress levels of their parents. (2) Methods: We enrolled 178 Italian adolescents aged between 12–18 that were referred to the Child Neuropsychiatry Unit of the University Hospital of Salerno with different neuropsychiatric diagnoses. Two standardized questionnaires were provided to all parents for the assessment of parental stress (PSI-Parenting Stress Index-Short Form) and the emotional/behavioral problems of their children (Child Behaviour Check List). The data collected from questionnaires administered during the six months preceding the pandemic, as is our usual clinical practice, were compared to those recorded during the pandemic. (3) Results: The statistical comparison of PSI and CBCL scores before/during the pandemic showed a statistically significant increase in all subscales in the total sample. The correlation analysis highlighted a significant positive relationship between Parental Stress and Internalizing/Externalizing symptoms of adolescent patients. Age and gender did not significantly affect CBCL and PSI scores, while the type of diagnosis could affect behavioral symptoms and parental stress. (4) Conclusions: our study suggests that the lockdown and the containment measures adopted during the COVID-19 pandemic could have aggravated the emotional/behavioral symptoms of adolescents with neuropsychiatric disorders and the stress of their parents. Further studies should be conducted in order to monitor the evolution of these aspects over time.