The article is devoted to the analysis of the author’s research: focus groups with parents of children gadget users with access to the Internet (n = 6), the questionnaire survey among adolescents (n = 303) and parents (n = 326) about media consumption, and information security legislation. The new trends and risks in the development of modern child-parent relationships: gadget and Internet addiction, conflicts, decreasing parental control and authority, alternative interaction forms, dangerous content and the consequences of its consumption were identified. The research objectives are to identify the main areas and problems of interaction between parents and children with IT-technology products, zones and consequences of media influence on the parent-child relations and classification of strategies for the Internet safety of children. As a result, the main types of strategies for prevention and liquidation online risks: external (institutional) and internal (intrafamily, intrapersonal), restricting, training and replacing are typologized and presented.