Generative AI (GenAI) platforms, such as ChatGPT, have gained popularity among the public due to their ease of access, use, and support of educational and creative activities. Despite these benefits, GenAI poses unique risks for families, such as lacking sufficient safeguards tailored to protect children under 13 years of age and not offering parental control features. This study explores how parents mediate their children's use of GenAI and the factors/processes that shape this mediation. Through analyzing semi-structured interviews with 12 families, we identified ways in which families used and mediated GenAI and factors that influenced parents' GenAI mediation strategies. We contextualize our findings with a modified model of family mediation strategies, drawing from previous family media and mediation frameworks. We provide insights for future research on Family-GenAI interactions and highlight the need for more robust protective measures on GenAI platforms for families.