Emergency crews responded to the scene at about 10:20 a.m., and found two people who had been injured, Capt. David Sadecki said.
The market is at 2886 Mission Drive and the gazebo is located on the store's patio.
A 3-year-old boy was momentarily trapped under the vehicle, but his father was able to pull him from underneath, Sadecki said.
Both were taken to Santa Barbara Cottage Hospital with minor injuries.
The vehicle was driven by a 73-year-old woman, and she and her husband sustained no injuries as a result of the accident, Sadecki said.
The woman was pulling into the parking lot and preparing to park when she stepped on the accelerator instead of the brake, said Officer John Ortega, public information officer for the California Highway Patrol.
Her car jumped a curb, went through a fence and struck the father and son in the gazebo before coming to rest, he said.