Child screen media use may cause family conflict, and risk factors for such conflict are not well characterized. This study examined risk factors of persistent requesting to use screen media among preschool-age children, focusing on parent-reported characteristics of parent and child screen media use. Data were collected through an online survey completed in 2017 by a nationally recruited sample of 383 parents of 2–5-year-old children. Parents reported on their child's and their own screen media use, household/sociodemographic measures, and child requests to use screen media. Persistent requesting was defined as exhibiting “bothersome” or “very bothersome” behaviors to use screen media. Poisson regression with robust standard errors computed the prevalence risk ratio of persistent requests on parent and child screen media use characteristics, adjusted for household and sociodemographic characteristics. Overall, based on parents' reports, 28.7% of children exhibited persistent requesting, which was often accompanied by whining, crying, gesturing, or physically taking a device. In an adjusted regression model, higher amounts of parental time spent using social media, but not parental time spent using other screen media, was associated with a greater prevalence of children's persistent requests. In latter models, children's use of smartphones and engagement with online videos were independently related to persistent requests. Across all models, children's total quantity of screen media use was unrelated to persistent requests. Practitioners advising families on managing conflict around child screen media use should consider characteristics of both child and parent screen media use.