The narrator of "Who's Irish?" is a sixty-eight year old Chinese grandmother who lives with her daughter, son-in-law, and granddaughter. Years earlier, she came to America with nothing and was unable to speak English, yet she and her husband worked hard and created a comfortable life for themselves and their daughter.
To determine whether the grandmother is a central or peripheral narrator, you should consider the conflict. The grandmother is central to the ongoing conflict with her daughter, her son-in-law, and Sophie. She also is central to the resolution of the story, when she goes to live with Bess. Since these conflicts are varied and center around the grandmother herself, she is the central narrator in this story. If you remove her from the story, the conflict would be dramatically altered.
To a great extent, the grandmother is a reliable narrator. She seemingly presents the conflict fairly objectively, even "admitting" to the reader that she spanks Sophie without telling her daughter and son-in-law. She admits to poking Sophie with the stick and doesn't shy away from her feelings about her Irish in-laws. Yet through her various comments on cultural differences, we also must see that the grandmother has prejudices of her own. Consider this comment early in the story:
Even the black people doing better these days, some of them live so fancy, you'd be surprised. Why the Shea family have so much trouble?
The grandmother has low expectations of the Black community and is even "surprised" to realize that some Black families are doing well. She is also prejudiced against the Irish, saying that they have bland food and bland names. The grandmother expects everyone to act according to specified roles and is frustrated when people like her son-in-law don't measure up to her expectations. So while she is an overall reliable narrator, we must also consider that her commentary is filtered through her own notable biases.
The grandmother builds an intimacy with readers through her frustrations. She adores her daughter and wants the best for her. Yet she is also caught in a chasm of generational differences that creates strife between herself and her "wild" granddaughter who cannot even keep her clothes on. The grandmother is forbidden from spanking Sophie, yet Sophie's mother doesn't do much to discipline her daughter, either. The grandmother wants her son-in-law to take some responsibility and is frustrated by his comments that he cannot babysit Sophie because it isn't "manly." In these conflicts, readers realize the humanity of the narrator as she struggles to maintain the relationships which she holds so dear. Despite her best efforts to be the mother she thinks her daughter needs, she still falls short and is sent to live elsewhere. In her shortcomings, readers can empathize with the delicate balance of loving people and allowing them to make their own choices, even when we don't personally agree with those choices.
Thus, the story is told to readers in hopes of finding the sympathetic ear which the narrator's daughter never offered. She doesn't blame her daughter for making her move out, understanding that divorce is a "terrible idea." She simply comes to the reader to share her experience as a means of embracing the next phase of her life.