This study aimed to investigate the relationship between media addiction levels among early school-aged children and their health status, self-esteem, and their fathers’ parenting practices. Therefore, we analyzed the data from a total of 1149 fathers and children from the 10th year (2017) survey of the Panel Study on Korean Children (PSKC), by the Korean Children’s Panel Research Institute. Specifically, a multinomial logistic analysis was conducted to identify the factors affecting children’s media addiction. The media addiction levels were 68.8%, 24.9%, and 6.3%, in general, high-risk, and potential-risk users, respectively. When media addiction levels were used as the reference group for general users, the pattern observed in the data revealed a direct positive association between media use time and the probability of becoming a potential-risk user (79.4%, OR = 1.79, 95% CI: 1.20, 2.68). Furthermore, when general users were used as the reference group, the male gender accounted for the majority of high-risk users, at 99% (OR = 1.99, 95% CI: 1.46, 2.71). Within this group, the children’s body mass index (BMI) was high (5%, OR = 1.05, 95% CI: 1.00, 1.11), their media use time increased by 1 h a day (145%, OR = 2.45, 95% CI: 1.93, 3.11), their self-esteem decreased (32%, OR = 0.68, 95% CI: 0.49, 0.95), the fathers’ authoritative parenting practices were low (37%, OR = 0.63, 95% CI: 0.43, 0.92), and the permissive parenting practices were high (92%, OR = 1.92, 95% CI: 1.09, 3.37). Therefore, the results of this study highlight children’s media use time and the risk factors related to high BMIs in order to prevent media addiction among early school-aged children. Our findings also suggest appropriate parenting practices and highlight the need to strengthen children’s self-esteem.