Mental Health Justice and Parity Act of 2022 This bill creates a grant program for mental health first responder units. It also eliminates a provision that permits nonfederal government health insurance plans that are self-funded to opt out of requirements to provide parity between coverage of medical services and mental health services. Specifically, the Substance Abuse and Mental Health Services Administration (SAMHSA) must award grants to states and local governments to train and dispatch mental health professionals to respond, instead of law enforcement officers, to emergencies that involve people with behavioral health needs. SAMHSA must manage the program in consultation with the Department of Justice (DOJ). SAMHSA may cancel grants that increase incarceration or institutionalization. Grantees must use funds for purposes including de-escalation and anti-racism training. The Department of Health and Human Services and the DOJ must evaluate this program.