Large Language Models (LLMs) have gained traction in educational settings, often framed as virtual tutors or teaching assistants. Following early skepticism and bans, many schools and universities have begun integrating these systems into curricula. Yet decisions about whether and how to deploy LLM-based tools are frequently made without systematic engagement with the full range of stakeholders they affect. In this paper, we argue that understanding stakeholder perceptions of LLM-based systems in the classroom is not a matter of measuring approval or acceptance, but of identifying whose concerns are surfaced, in which contexts, and with what implications for responsible design and governance. We introduce Contextualized Perceptions for the Adoption of LLMs in Education (Co-PALE), a stakeholder-first framework that connects educational context, responsible AI principles, and categories of perception to support more deliberate decision-making about the adoption of LLM-based tools. We ground Co-PALE through a targeted analysis of prior work to diagnose recurring gaps in how stakeholder perceptions are studied, and through contextually distinct educational scenarios that illustrate how the same technology raises different concerns for different stakeholders. We further examine how university faculty and K--12 parents make sense of the framework through focus groups, using their reflections to surface tensions and uncertainties. Co-PALE supports more systematic reasoning about whether, where, and for whom LLM-based tools should be deployed in education.