Well, if you consider how the story is meant to be, we've only played Parasite Eve 2 & 3. It's a bit hard to explain, but it should make sense (and give some glimpses of why there may not be another Parasite Eve per se, but why there can be more games with the main protagonist)

Thing is, the real first Parasite Eve is a Japanese horror movie based from a novel. It's not full of that much action, but it's pretty full of gore, as far as I can gather. The main, key elements (mitochondria, evolution and the rebellion of a body organ) emerge from the movie, and many of the events in Aya's past refer to the movie actually.

Parasite Eve (the videogame) is an adaptation of the game and the virtual sequel. Some of the original movie's main story was retconned to allow Aya to fit, but the plot of the movie fits quite well, and those who actually saw the original movie may recognize the elements drawn from it.

Parasite Eve 2 is the direct sequel to the videogame, but it departs a bit from the movie origins. It still uses the mitochondrial creatures, and it still refers to "Eve" and the New York incident, but it makes few or no references to the movie whatsoever (as far as I understand)

The so-called "Parasite Eve 3" is really a completely different game, most appropriately called "The Third Birthday", even though it ties a bit (or so Wikipedia claims) to the original two games. It's not meant to be a real PE3, since it doesn't contain any instances of mitochondrial creatures, but it works on how much the survival horror genre has evolved through the years (as well as the 3rd person shooter genre) As well, The Third Birthday has not been released as of yet; it will be released eventually for the PSP since it was originally expected as a cellphone release (much like the sequel to FFIV), but America doesn't have a cellphone system that could even duplicate or emulate the DoCoMo, and American cellphone companies are too hard to deal with (imagine that T3B was released only with...say, Sprint, or AT&T, or T-Mobile)

Parasite Eve was quite original in that it blended the developing survival horror genre with RPG elements. Most of the developments in PE have bled into other games (such as the weapon upgrade, which was essentially recovered and updated for Resident Evil 4), because the story is very solid (it is based on the book, but it blends the new features excellently with the old), it gives the feeling of horror and suspense without scaring (because you usually got the alert when the screen "froze" and you entered combat mode), and because the main character has powers. Mostly, the last one: while other games had heroes that barely had any power to survive and depend greatly on scavenging resources, Aya possesses virtually the same kind of weapon her enemies have, and to a much more refined degree.

I'd personally would love to replay the old game (never got to finish it, nor saw the real ending), and I look forward for 3rd Birthday to see how the system has changed (as far as I've heard, Aya will retain her powers or somehow get a new use for them, and the system will be closer to a third person shooter as other survival horror games)