The incident happened just after 2 p.m. Thursday at Holly Park in the city's Bernal Heights District. By Thursday evening, a memorial of flowers had been set up at the park where the woman was struck.
A witness said the truck was driving across the grass, which was busy at the time, and ran over the woman who was sunbathing with her child and dog nearby.
The woman was transported to San Francisco General Hospital with life-threatening injuries. The child and dog were not hurt.
The driver of a green Parks Department vehicle took off after the incident but was later found and detained by police at Crescent and Agnon avenues. The driver has been arrested on suspicion of felony hit-and-run.
Neighbors say that parks vehicles are always there, and it's a big problem with people driving on the grass.
"Bound to happen, I'm surprised that we haven't had more dogs run over because some people let their dogs run loose," neighbor Patricia Leach said.
Leach has lived there since the 1950s. She said she hopes there will be some sort of policy change in the future because the park is too busy to have vehicles driving on it.
Supervisor David Campos said he was not aware of the concern but promises a thorough investigation.
"At the end of the day, we want to make sure that our citizens are safe and that they can enjoy parks and rec space in a safe manner," he said.