Generative AI (GAI) technologies are quickly reshaping the educational landscape. As adoption accelerates, understanding how students and educators perceive these tools is essential. This study presents one of the most comprehensive analyses to date of stakeholder discourse dynamics on GAI in education using social media data. Our dataset includes 1,199 Reddit posts and 13,959 corresponding top-level comments. We apply sentiment analysis, topic modeling, and author classification. To support this, we propose and validate a modular framework that leverages prompt-based large language models (LLMs) for analysis of online social discourse, and we evaluate this framework against classical natural language processing (NLP) models. Our GPT-4o pipeline consistently outperforms prior approaches across all tasks. For example, it achieved 90.6% accuracy in sentiment analysis against gold-standard human annotations. Topic extraction uncovered 12 latent topics in the public discourse with varying sentiment and author distributions. Teachers and students convey optimism about GAI's potential for personalized learning and productivity in higher education. However, key differences emerged: students often voice distress over false accusations of cheating by AI detectors, while teachers generally express concern about job security, academic integrity, and institutional pressures to adopt GAI tools. These contrasting perspectives highlight the tension between innovation and oversight in GAI-enabled learning environments. Our findings suggest a need for clearer institutional policies, more transparent GAI integration practices, and support mechanisms for both educators and students. More broadly, this study demonstrates the potential of LLM-based frameworks for modeling stakeholder discourse within online communities.