Preschool children are immersed in screen media, yet little study has been conducted on screen addiction among them. This study aimed to investigate the relationship between parental screen addiction and young children’s screen addiction and to verify factors that mediate this relationship. A total of 477 parents of kindergarteners (3–6 years old) were recruited via six kindergartens in Henan province, China. They completed the Problematic Media Use Measure Short Form of Children, the Self-Rating Anxiety Scale, the Child–Parent Relationship Scale, and the Parental Screen Addiction Scale. The results showed that the relationships between each pair of parental screen addiction, parental anxiety, and children’s screen addiction were significantly positive, but the parent–child relationship was negatively correlated with the other variables. Parental screen addiction can directly and indirectly affect children’s screen addiction through parental anxiety and the parent–child relationship. The findings contribute to the development of family protection mechanisms against screen addiction in children.