Based on data from the 2010–2011 Korean National Health and Nutrition Examination Survey, we investigated correlations between micronutrients in the diet of family members and the possible risk factors for children and adolescents consuming an inadequate diet. We examined two-generation households with children aged 2–18 years. The quality of the family diet with regard to the following nine nutrients (protein, calcium, phosphorous, iron, vitamin A, vitamin B1, vitamin B2, niacin, and vitamin C) was assessed based on the Index of Nutritional Quality. Correlations between quality of diet and selected variables were analyzed using the Statistical Analysis for Genetic Epidemiology software, and those between diet quality and potential risk factors for poor diet in offspring were analyzed using multinomial logistic regression. Overall, calcium was the most commonly under-consumed micronutrient. More than half of sons and daughters showed insufficient vitamin A, vitamin C, and iron intake, and both mothers and fathers showed insufficiency with respect to vitamin A, vitamin B2, and vitamin C. The correlation between a poor diet in parents and that in offspring was 0.17 (p < 0.0001), and this correlation coefficient was higher between mothers and offspring than between fathers and offspring. Additionally, eating breakfast provided a significant protective effect against the risk of poor nutrition in offspring, even after adjusting for covariates. Our results add to evidence indicating that children should be encouraged to eat breakfast to improve the quality of their diet.