What is the difference between a funeral home, mortuary and funeral parlor?

To start, these are the definitions of each of those terms from Merriam-Webster: 

Funeral Home

an establishment with facilities for the preparation of the dead for burial or cremation, for the viewing of the body, and for funerals

Mortuary

a place in which dead bodies are kept until burial

Funeral Parlor 

a place where a dead person is prepared to be buried or burned, and where relatives and friends can see the body

A mortuary is often more focused on providing burial and cremation services over funeral services. This term is commonly used for facilities designed specifically for the preparation of bodies before the transport to a final resting place. This is similar to a morgue. A morgue temporarily stores a body before being transferred to a funeral home or mortuary for final preparation before burial or cremation. 

Before the civil war, funerals were often held in the family home in the room called a parlor. After funeral homes were becoming increasingly common, it was natural for them to be named “funeral parlor”. The name funeral parlor also originated from the room where a viewing was held- a parlor. In the early days after the Civil War, most undertakers would use their homes to run their funeral business from. This is most likely where the name funeral home comes from. 

While it is common for these terms to be used interchangeably, each term has a specific meaning in the funeral industry.