In order to keep this relatively short, I've gotta give a cliff's notes of the first two games.
In the first game, Aya first learns about her abilities as an entity named Mitochondria Eve immolates hundreds of people at an opera and, when Aya tries to stop her, inadvertently unseals her own abilities. It turns out that Eve is a form of lifeform based on Mitochondrial evolution and intends to kill every existing form of life in favor of producing superior organisms that can manipulate their own DNA. Due to a mutation she has from a cornea transplant, Aya uniquely is impervious to Eve's induced mutations and has similar abilities she can manifest, and she winds up fighting Eve to the death (and subsequently, her offspring) in NY, successfully killing it and bringing an end to the conflict.
The second game more directly involves Aya because of what transpires. Aya winds up roped into dealing with a series of NMCs (neo mitochondrial creatures; the mutated shit that Eve initially created), and as the plot moves forward it becomes apparent that someone is
creating unique versions of NMCs for some nefarious purpose. Aya digs deeper in, and, if you have picked up all the clues to the right ending, finds out that there was a major conspiracy to spread a retrovirus to induce NMC mutation globally. Central to this was a young clone of Aya, raised in an environment to boost her survival instincts, but she had gotten so good at protecting herself that the group that intended to make this plan happen couldn't get to her. In the end, Aya kills an ultimate lifeform or two, helps expose members of the government who had been involved in this plan, and gotten a presidential commendation, as well as an adoptive younger sister in the form of aforementioned clone.
Aya's whole thing is that she's an ex-cop turned investigator with freaky genetic powers that let her burn things alive via spontaneous combustion or poison them by making their cells die. The games establish that a big reason for the differences between her powers in the two games is that Aya didn't use her powers very much between them, causing them to decay (and evolve, as it turns out, when she starts using them again).
Parasite Eve as a whole is a very character-driven piece, with its protagonist, Aya Brea, being a bit of an outlier as far as video game heroines go. She's intelligent, sarcastic, and has a sense of dark humor, but she's also someone who's had to deal with a lot of terrible implications brought about by her abilities. She faces down the constant prospect that if she loses it, she could kill a lot of people or worse, turn into something like Mitochondria Eve. Both games are actually based on a book series, which is actually referenced in both games as being canon with them.
By the time
The Third Birthday was being produced, Squeenix had
lost the
Parasite Eve license and apparently didn't want to renew it. They went about making a game without it, which is about as terrible an idea as you might think. Worse, they assigned FFXIII's team to it, which is like taking a bad idea and then convincing Kengle to have sex with it. The result was a game that wasn't necessarily damned from the start, but quickly got that way anyway.
All the characters that are in
The Third Birthday and happened to appear in previous games either die horribly soon or are so completely unlike they are supposed to be that you wonder how its possible to be that out-of-touch. To start with, Aya herself is quiet, non-confrontational, and weak - completely unlike the character we actually know from the previous games.
In fact, there's a reason for this - Aya herself isn't Aya at all, but in fact, Eve, her adoptive sister from the second game, who happens to be in her sister's body and yes, we're going into an area just as creepy as you think this might be. In the canon for
The Third Birthday, Aya's wedding to PE2's Kyle Madigan (something I could have actually seen happen) was apparently attacked by a SWAT team for reasons never explained by the plot. Eve did....
Something that caused her to inhabit Aya's body and caused the emergence of the Twisted, the enemies of
The Third Birthday.
The plot proceeds forwards, making virtually no sense, and featuring time travel because
of course it does. During this time you see that Aya is much more obviously sexualized, with the camera often focusing on her ass and her clothes being torn to shreds during combat. So we literally have the spirited, deep, intelligent character from previous games replaced with a meek kittenish upstart who is packed to the gills with fanservice. Whilst the first two games had their share of nudity, it was more-or-less there to get a point across; in
The Third Birthday, you could legitimately make the case that there was some for-realz misogyny going on here.
Oh, but Aya isn't the only character to get this treatment, oh no. You get Maeda, one of the most likable, affable characters from PE1, turned into this sick sexual predator in
The Third Birthday. Kyle Madigan has no real impact on the plot besides being the guy Aya was going to marry and one of the bosses after he turns into a giant skyscraper-sized monster. Jodie Bouquet, the gossipy weapon shop girl from PE2 shows up in a cutscene, only to be torn in half by one of the Twisted, and though it's not clear whether or not it's actually her, I choose to believe it is given the rampant fucking contempt the game has for its orginal fanbase.
The story is even worse. I'm not fucking kidding.
It all comes to a head later where the real nature of Aya and Eve in The Third Birthday is made clear - and for one, brief, glorious moment, we get old Aya back, the snarky badass. Only to have to murder her to make the Twisted never happen and Aya's 16-year-old clone sister to occupy her body and go on to marry Aya's husband with him nonethewiser and I have to stop here or I'm going to go to Square-Enix and take someone hostage. I want to point out that
I played through Revolution 60 without complaint, so you have a general grasp of how bad this story is.
For more information, consult Games I Fucking Hate's review which explains it quite nicely: