How are people hearing the story? The person telling people about mom's goof should stop. If it's the daughter, then she should apologize for spreading it.
Sometimes you apologize for things you aren't at fault for. When you want to express sympathy, for example. Here I think the daughter would be doing that plus expressing that she felt bad that mom was being chastised on *her behalf*. Hence, Prudie's advice that the daughter tell others to knock it off.