Background: The effect of stress on mental health has been increasingly acknowledged. Drinking habits are closely inter-related with stress and each affects the other. However, only limited studies addressed the effects of alcohol consumption on family members apart from spouses. The purpose of this study is to better understand the relationship between parent drinking frequency and their children’s self-reported stress. Methods: Data was collected from the Korean National Health and Nutrition Examination Survey (K-NHANES) conducted during 2007–2016. Respondents were divided into three groups: children (n = 3796), maternal (n = 22,418), and paternal (n = 16,437). After merging the children and parents data sets, we identified the final study population of 3017 and performed binary logistic regression. Results: We found that the odds of high stress cognition was 1.58-fold higher for children who have heavy drinking mother (95% CI: 1.14–2.19) and 1.45-fold higher for those who have heavy drinking father (95% CI: 1.06–1.99). In a subgroup analysis, children whose household income level was Q1 and maternal occupation was white collar showed a statistically significant association of high stress with parental drinking frequency. Conclusions: Parental drinking frequency negatively impacts stress in the children of drinkers. We suggest providing support care for children in vulnerable environments to improve their stress levels.