The officer was identified as Salvador Sanchez of the LAPD's Southwest Division, the department announced Wednesday.
Sanchez shot at the family after an argument ensued at the Costco in Corona on Friday night, a Corona Police spokesperson said at a press conference.
Police said Sanchez was assaulted "without provocation" by a 32-year-old man while he was holding his young child. The officer allegedly then opened fire and struck the man, identified as Riverside resident Kenneth French, and his parents.
Sanchez also suffered minor injuries in the incident, the LAPD said in a statement after the shooting.
Sanchez's attorney, David Winslow, told The Associated Press on Monday that Sanchez was struck from behind, without a word being spoken, as he fed his son samples of teriyaki chicken.
Sanchez believed that his and his son's life were in danger, Winslow said.
The family's attorney, Dale Galipo, told AP on Tuesday that he wasn't aware whether there was an exchange between Sanchez and French, describing the encounter as an "open-handed push or slap" rather than "an attack."
"It certainly does not justify killing someone," Galipo said.
A cousin of French described him to AP as a gentle man who was nonverbal because of a mental condition. The family believes he suffered from schizophrenia, but he had been taken off his medication due to other health complications, which may have affected his behavior that night, Galipo said.
The LAPD is conducting an internal investigation into the shooting, according to the release. A spokeswoman for the department declined to provide additional comment to ABC News.
As of earlier this week, French's mother remained unconscious and in critical condition, while his father was in serious condition but was conscious, Galipo said.
The current state of their conditions was not immediately clear.
ABC News' Soo Youn contributed to this report.