Three students are in critical condition but stable, Gulf Shores city spokesman Grant Brown said. The victims' ages range from 12 to 17.
Brown said a 73-year-old man was driving the SUV, which accelerated before it hit the Gulf Shores High School marching band. It was not immediately clear why the driver accelerated, but authorities do not believe the crash was intentional.
"Every indication is that it was a tragic accident," police Chief Ed Delmore said.
Senior Jeremy Koonce told CNN affiliate WPMI that he was in the section of woodwind players that was hit, but someone pushed him out of the way.
"A couple people got rolled under the actual truck, and I saw people roll out from underneath it, and they were like clutching themselves in pain," Koonce told the station.
As the children were treated and rushed to ambulances, dented musical instruments and debris were scattered on the street.
One of the students was released from the hospital, the school system said Tuesday afternoon on its Facebook page. Cindy Diehl indicated to WPMI the student was her 15-year-old daughter, who had a bruised pelvic bone and road rash.
"... but I think she's going to do pretty good," said Diehl.
Six of those injured are middle school students; six are high school students, Baldwin County Public Schools
said in a statement.
Brown said there were no signs that drugs or alcohol were involved. Delmore told reporters at a second news conference that the driver had given blood samples voluntarily.
The chief said there is at least one video showing the accident, and police also hope to retrieve information about the vehicle's speed and movement from an computer on the SUV.
Several emergency agencies scrambled to the coastal community to help with what fire Battalion Chief Kevin Tolbert called a "mass trauma incident."
Tolbert said the SUV ended up on top of some of the children.
A crisis counselor was at a local high school to comfort students and parents who witnessed the crash, said Angie Swiger, a member of the Baldwin County Board of Education.
"It's going to be a very difficult day," she said. "It is a tight-knit community. We just ask for everyone's thoughts and prayers."
The vehicle was representing the Military Officers Association of America. The organization said it was still learning the facts.
"We're concerned about those who have been injured and our hearts and prayers go out to them and their families as they deal with this unfortunate mishap," the organization said.
Brown said the band was at the head of the parade and the SUV was behind the 100 or so members. The band marched onto a highway to begin the parade and the truck followed.
"And then something went terribly wrong," Brown said.