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Epidemiological studies indicate that children of parents with mental health problems are at an increased risk of developing anxiety disorders. Few studies have investigated this relationship in young adults. Participants were from the Raine Study, which is a multi-generational birth cohort study in Australia. Maternal anxiety and depression in late childhood were assessed using the Depression, Anxiety, and Stress Scale (DASS-42), and paternal lifetime mental health problems were assessed using a self-reported questionnaire. The short form of DASS-42 (DASS-21) was used to assess anxiety symptoms among offspring at age 20. Negative binomial regression model was used to quantify the association. Data were available for 1,220 mother-offspring and 1,190 father-offspring pairs. After adjusting for potential confounders, we found an increased risk of anxiety in young adult offspring exposed to maternal anxiety in late childhood and paternal lifetime mental health problems. However, we observed no increased risks of anxiety in offspring exposed to maternal depressive symptoms. Our sensitivity analysis based on the log-binomial model (binary outcome) as well as the linear model (log-transformed data) confirmed the robustness of the main results. Our findings suggest there can be value to consider and apply targeted screening and intervention programs of anxiety in the young adult offspring with parental mental health problems.