Piper is the daughter of Aphrodite, so it makes sense that much of her story is about love of all sorts. You'd think her mom would go a little bit easier on her though, because love doesn't make Piper happy. Instead, love kicks Piper's butt up one perfumed pathway and down the next.
Piper's most important relationship is with her dad, Tristan McClean—she adores the guy, but she can't get him to pay much attention to her, despite stealing BMWs (which, admittedly, doesn't seem like the best way to get attention). Her next most important relationship is with Jason Grace. He becomes her boyfriend at Wilderness School after she works hard to overcome his shyness and finally, finally gets "the big dope to kiss her" (3.5). And then his memory is wiped and she's "stuck in the worst 'do over' of all time" (3.6). Even worse, she eventually discovers that her own memories of Jason are just an illusion—they never really were boyfriend and girlfriend at all. Ugh.
As if that's not terrible enough, through much of the book, Piper is violently torn between these two already painful loves. Her father is captured by the giant Enceladus, who tells her that he'll kill her dad if she doesn't betray her friends (including that big dope Jason). She gets all the fear and pain of love without any of the good stuff. It's a surprise she doesn't deck her mom when she sees her.
But is Piper really so unlucky in love? Her mother, Aphrodite, tells her she sees "possibilities much more vividly than others. You see what could be" (39.30). Rather than seeing Piper's relationship with Jason as a do-over, you could see it as a do-once. She dreamed or imagined that he kissed her, but of course that kiss "never really happened" (22.72). Then she sets out to make sure it does happen—and while she hasn't quite succeeded by the end of the book, it's clear she will eventually. She made a wish for love, and that wish is going to come true.
Her relationship with her father is less straightforward, though it's still not too hard to see it as a kind of wish-fulfillment. Her dad is kind of a jerk—even when she begs him to spend more time with her, he just sends her off with his stupid assistant, Jane, who puts her in yet another school for troubled kids.
Piper never expresses this directly, but we think she must be pretty angry with her dad. She definitely makes her anger with Jane clear, at one point even praying that the assistant would fall over and knock herself out (9.33).
And then what happens? It turns out that Jane is the pawn of Medea, and is in fact every bit as evil as Piper always thought she was. Her father is thoroughly punished for ignoring her by a gruesome giant, and Piper herself saves him, forcing him to tearfully admit that "you're a real hero, not like me" (45.25). Best of all, she uses a potion to erase his memory, so that he suffers no permanent damage; it's only Piper who knows that he suffered and that he needed her. She has her all the satisfaction of revenge and I-told-you-so without the guilt that would result if he'd actually been hurt. The memory loss potion from this perspective can be seen as affecting not only her dad, who forgets he was kidnapped, but Piper herself, who forgets that she maybe, deep down, in some recess of her heart, wanted him to suffer just a bit.
The book, then, can be seen as one long wish-fulfilling daydream for Piper, who gets the boy she wanted and shows for sure that her dad needs her. Maybe her mom loves her after all.