Was there a Parselmouth before Salazar Slytherin?

Yes, Herpo the Foul is a Parselmouth who pre-dates Slytherin.

From Fantastic Beasts and Where to Find Them (emphasis mine):

The first recorded Basilisk was bred by Herpo the Foul, a Greek Dark wizard and Parselmouth, who discovered after much experimentation that a chicken egg hatched beneath a toad would produce a gigantic serpent of extraordinarily dangerous powers.

According to Professor Binns in Chamber of Secrets:

You all know, of course, that Hogwarts was founded over a thousand years ago - the precise date is uncertain - by the four greatest witches and wizards of the age.

The books are set in the 1990s, which means Slytherin would be active around the tenth century. This means he almost certainly didn’t overlap Herpo the Foul, and so Foul is a Parselmouth who pre-dates Slytherin.

Edit: As Anthony Grist points out in the comments, this only says Greek, not Ancient Greek. I read Ancient Greek in the HP Wikia entry, and conflated the two. This is confirmed in his chocolate frog card on Pottermore:

Herpo the Foul was an Ancient Greek wizard and the first known creator of the Basilisk.

---

Note that most known Parselmouths in the HP universe are connected to Slytherin. (The Gaunts, Voldemort, Harry by proxy to Voldemort.) Indeed, JKR confirmed in a 2007 interview that the ability has come down via Slytherin (emphasis mine):

Q: Since Ron is able to speak Parseltongue in the last book, does that mean that parseltongue is a language that most witches and wizards can learn or must a person be born with some ability to speak Parseltongue.

JKR: I don't see it really as a language you can learn. So few people speak it that who would teach you? This is a weird ability passed down through the Slytherin blood line.