Children are increasingly using technologies powered by Artificial Intelligence (AI). However, there are growing concerns about privacy risks, particularly for children. Although existing privacy regulations require companies and organizations to implement protections, doing so can be challenging in practice. To address this challenge, this article proposes a framework based on Privacy-by-Design (PbD), which guides designers and developers to take on a proactive and risk-averse approach to technology design. Our framework includes principles from several privacy regulations, such as the General Data Protection Regulation (GDPR) from the European Union, the Personal Information Protection and Electronic Documents Act (PIPEDA) from Canada, and the Children's Online Privacy Protection Act (COPPA) from the United States. We map these principles to various stages of applications that use Large Language Models (LLMs), including data collection, model training, operational monitoring, and ongoing validation. For each stage, we discuss the operational controls found in the recent academic literature to help AI service providers and developers reduce privacy risks while meeting legal standards. In addition, the framework includes design guidelines for children, drawing from the United Nations Convention on the Rights of the Child (UNCRC), the UK's Age-Appropriate Design Code (AADC), and recent academic research. To demonstrate how this framework can be applied in practice, we present a case study of an LLM-based educational tutor for children under 13. Through our analysis and the case study, we show that by using data protection strategies such as technical and organizational controls and making age-appropriate design decisions throughout the LLM life cycle, we can support the development of AI applications for children that provide privacy protections and comply with legal requirements.