This is the short story "Kid Brother" by Isaac Asimov which you may have read in the collection Gold.
A (slightly questionable) version of the story can be found here and it was pointed out by @DavidW that you can borrow Gold from the Internet Archive. The plot matches your summary almost exactly.
The parents have a son, Charlie, but are unable to have another child. However, the mother doesn't want Charlie to grow up as an only child and pushes to get a "sibling-robot" for him, which the father reluctantly agrees to. Despite the father's initial hesitation, the robot becomes accepted and integrated into their family but is especially favored by the mother.
One day, while the father is at work, their home "microfusion unit" malfunctions, causing a fire. Rushing home, the father finds the wife sobbing on the lawn and repeating, "I couldn't save them both." He attempts to comfort her by telling her that the insurance will cover another robot and then goes to check on his son, lying nearby and surrounded by a crowd.
Upon inquiring about the state of his son, one of the bystanders says, "Maybe he can be fixed up," and the father sees that the mother saved the robot, not Charlie.