Dubliners is a compliation of short stories by James Joyce. Each story has its own plot and reflects an issue that Joyce had become familiar with whilst growing up in Dublin, Ireland.
Polly, from The Boarding House, lives with her mother, a woman of questionable morals but a singular purpose - to improve her daughter's chances of a brighter future than her own. Mrs Mooney realizes the bad influence her husband has on his children, especially when he chases her with a meat cleaver. She manages to arrange a separation through the church - divorce was illegal at the time for Catholics. Her boarding house provides sufficiently for the family and Mrs Mooney even removes Polly from her typist job due to her father's uninvited interference.
Despite the "catholic" influence, Polly is given "the run of the young men" as her mother watches from the sidelines, waiting for someone of some worth but "none of them meant business" until Bob Doran. It is Polly herself who almost seduces Doran but her mother's encouragement, whilst subtle, is there and Doran is "persuaded" or rather, manipulated, into agreeing to marry Polly - to save her apparent reputation. Polly then is most certainly influenced by her mother and, seemingly, saved from her father's abusive tendencies.
Eveline, from Eveline, fondly remembers her childhood, much like any other, with a strict, temperamental father but nonetheless, her early childhood seems " to have been rather happy then." Unfortunately her mother dies and her father's violent nature intensifies and he threatens not only her brothers but her to the point that she has decided to leave "home." She knows her life has been hard and she has had to fulfill all her mother's duties but compared to the unknown life that awaits her with Frank, her "beau", maybe it's not so bad. Her father forbad her from having anything to do with Frank but now she intends to leave with him for Buenos Aires, such is the overwhelming influence of her father from which she wants to escape. "Escape!" Eveline remembers how hard her mother's life had been and how her father had abused her and made her miserable. This is the life she must escape from.
Hence Polly and Eveline are trying to improve their future potential to ensure that they do not suffer the fate of their mothers. Eveline, however, is not strong enough to do it and ultimately stays behind. Both women do feel affection for Bob Doran and Frank respectively, but Eveline does not know if she is escaping or whether Frank "will drown her."
Jimmy, from After the Race, has a starkly different start to life. His father is a successful "merchant prince" and Jimmy is well-educated and privileged. He even gets "to see a little life." Jimmy does understand his father's efforts in securing his son's position ensuring he practises only "reasonable recklessness." His father is however proud of Jimmy's social skills and seems to place woth on his associations. Jimmy gets in to a card game and loses heavily but the value is perhaps lost on him as he can "cover up his folly." However, the reader is left wondering how, his father, perhaps allowing him too much space and not enough moral judgment , will feel if Jimmy's gambling goes to far.
All three characters then are influenced by the action of their parents in making life-changing decisions.
Please repost the second part of your question as, in terms of eNotes rules, only one question, or a second closely related question can be asked per post.