Sgt. Christopher Cook, an Arlington Police Department spokesman, referred all questions to Parker. Messages left for Parker by the Associated Press were not returned.
Carmen Brown told the Dallas Morning News that she was waiting in line to get on the ride when the accident happened. She said she saw the woman being strapped in.
"They didn't secure her right," she told the newspaper. "One of the employees from the park - one of the ladies - she asked her to click her more than once, and they were like, 'As long you heard it click, you're OK.' Everybody else is like, 'Click, click, click.' "
"Hers only clicked once. Hers was the only one that went down once, and she didn't feel safe, but they let her still get on the ride," Brown said.
Six Flags said the ride would be closed as the investigation continues. A concert scheduled for Saturday was canceled.
The Texas Giant is 14 stories high and has a drop of 79 degrees and a bank of 95 degrees. It can carry up to 24 riders. It opened in 1990 as an all-wooden coaster but underwent a $10 million renovation to install steel-hybrid rails and reopened in 2011.
When the car that the woman had been riding in returned to the loading zone, two people got out and were visibly upset, Rockwell resident John Putman told the Fort Worth Star-Telegram.
"They were screaming: 'My mom! My mom! Let us out, we need to go get her!' " Putman told the newspaper.
Six Flags Over Texas opened in 1961 and was the first amusement park in the Six Flags system. It is 17 miles west of downtown Dallas. The park's first fatality happened in 1999. A 28-year-old Arkansas woman drowned and 10 passengers were injured when a raft-like boat on the Roaring Rapids ride overturned in 2 to 3 feet of water.
A 2005 report to the U.S. Consumer Product Safety Commission estimated just over four people died annually on amusement rides from 1987 to 2002. The estimate includes both mobile amusement park rides and fixed-site rides.