Consensual Donation and Research Integrity Act of 2021 This bill requires entities that acquire or transfer human bodies or body parts for education, research, or the advancement of medical, dental, or mortuary science to register with the Department of Health and Human Services. Registered entities must comply with record-keeping, labeling, packaging, and disposition requirements. The registration and related requirements do not apply to the acquisition or transfer of human bodies or body parts for use in human transplantation.