Similar to her other novels, in Persuasion, Austen pinpoints differences in gender roles that were accepted in her society. However, she also points out that some people, regardless of gender, are incapable of performing their gender roles well.
We especially see this in how she characterizes Mary and Charles Musgrove. When their eldest son falls and dislocates his collar bone, Mary, although she is the mother, claims that she is too nervous and upset to be able to stay and nurse him. Mary would prefer to dine at Uppercross with Captain Wentworth. She is also extremely upset when Charles decides to leave the boy and dine at Uppercross the next night. When Anne reminds Mary that "nursing does not belong to a man; it is not his province. A sick child is always the mother's property: her own feelings generally make it so," Mary convinces Anne to stay alone and nurse him instead (Ch. 6). Hence, we see from this scene that both sisters are not equally cut out to play their feminine roles. Instead, only Anne plays her motherly, caring gender role well. We also see Charles playing his gender role by refusing to stay with the boy and preferring to be out in the world.
Although, in terms of parenting, Austen actually does switch the gender roles of Charles of Mary. Anne actually does believe that Charles is the better parent. Charles declares that he "could manage [the boys] very well, if it were not for Mary's interference," and Anne agrees. Anne believes that it is Mary who is spoiling the boys, despite the fact that Mary blames Charles. We see from this, that despite the scene above, Charles actually handles the woman's role of raising children much better than his wife does.
Hence, Austen does point out gender-specific roles. However, only Anne faithully fulfills her gender role, while the other characters perform their gender roles badly.