Officials say that the incident took place in the city of Gallup during the Intertribal Ceremonial Parade on Thursday.
Investigators say they first responded to a report of people drinking alcohol in a car along the parade route.
State police officers arrived at the scene and the driver drove away in the vehicle before the crash took place.
Court papers state that State police arrested Jeff Irving and a passenger after the SUV they were in crashed the vehicle on Historic Route 66 and 2nd Street in the city.
Video taken at the scene shows a group performing in the street when the crowd starts screaming and people rushing out of the way of the vehicle.
The parade was part of a days-long celebration of Native American culture and heritage in the city.
“The Navajo Nation stands with resilience against any acts of violence and sends prayers of protection to those affected,” said Navajo Nation Council Speaker Seth Damon in a statement.
“This was a traumatic and triggering event for many, especially for our youth, elders, and our veterans who acted quickly.”
And he added: “Hold Gallup in your prayers tonight as we come together in faith and strength for one another. May the Creator and Holy People bless you all tonight as we move forward together.”