Moving away from rigid, worksheet-heavy curriculums toward a true play-based learning model is academically the right choice for young children. However, executing that transition in a way that parents actually support is a massive challenge. When parents do not see physical paper coming home in their child's folder every afternoon, they immediately assume no learning is happening.
You have to educate your parents before you change the classroom. Host an evening workshop detailing the actual neuroscience of play. Explain how building with wooden blocks teaches early geometry and spatial awareness much better than tracing a shape on a piece of paper. More importantly, train your teachers to write incredibly descriptive daily reports. Instead of just checking a box that says "played outside," the report needs to say, "Sarah spent twenty minutes negotiating the rules of her own obstacle course, developing critical problem-solving and language skills." When parents can read exactly what their child is learning through play, their resistance disappears almost entirely.