The incident happened about 11:50 a.m. Saturday, KCBS/KCAL-TV, Los Angeles, reported.
Those injured were apparently riding horses, which got spooked by another horse running wild in the park.
"At 1150 a horse running loose on a trail in Equestrian Ctr property injured 4 adults &3 minors. All trans to local hospitals. LAPD handle," the Burbank Police Department tweeted. Police later said six people, not seven, were injured.
One of the riders who was injured told the TV station she was scared for her life.
"I just couldn't believe it," Katherine Novak said. "I was on the ground. And it was terrifying. There was this stampede running right at us."
A park employee said the horses involved in the incident did not belong to the park's equestrian center and that the riders were amateurs.