The amount of research on co-design in educational contexts has been increasing in recent years, providing insightful results. Despite this growth, the documentation of case studies has remained incomplete and inconsistent, hindering the validation of methods, results and overall maturity of this field of study. We systematically review 64 publications since 2005, with a focus on primary education, educational video games and STEM, to explore documentation trends and develop a taxonomy that outlines the most relevant co-design parameters (CDPs). Our mixed-method analysis facilitates the interpretation, discussion and implications of our findings, as they portray the current state of documentation practices and their shortcomings. The resulting taxonomy and CDPs serve as a guide to design and document future research more consistently, enabling replication and longitudinal studies and contributing directly to the maturity of education co-design.