The Mediterranean diet (MD) is known to be one of the healthiest dietary patterns. Despite the significance of a healthful diet during the early stage of life, data for young individuals indicate that nutrition problems are common. This cross-sectional study aimed to determine parental factors associated with MD adherence in children and adolescents living in the Mediterranean region in Croatia. In total, 2623 children aged 2 to 18 years and their parents participated in this study. Data were collected during the period from September 2021 to February 2022 by using an anonymous questionnaire. We used KIDMED and MEDAS questionnaires for assessing MD adherence in young individuals and their parents, respectively. To assess the association of children’s MD adherence categories with the parental predictors, we performed multivariate multinomial logistic regression. Results showed that the children of parents with a low MD adherence are much more likely to have poor MD adherence than good (OR = 47.54 (95% C.I 18.24, 123.87), p < 0.001) or average (OR = 5.64 (95% C.I 3.70, 8.6), p < 0.001) MD adherence. Further, children of fathers with higher BMI (OR = 1.035 (95% C.I 1.0, 1.071)) and those who do not live with both parents (OR = 1.703 (95% C.I 0.994, 2.916), p = 0.053) are more likely to have poor MD adherence than good MD adherence. These results indicate that interventions focusing on enhancing the quality of both parents’ diets could effectively improve their children’s eating habits.