Was Salazar Slytherin the first Parselmouth, or did somebody else use it before him?
I’ve read the books, but I can’t find any examples.
Yes, Herpo the Foul is a Parselmouth who pre-dates Slytherin.
From Fantastic Beasts and Where to Find Them (emphasis mine):
According to Professor Binns in Chamber of Secrets:
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.
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.
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.
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.
According to the Famous Wizard Cards seen in the video game "Harry Potter and the Chamber of Secrets", Paracelsus (1493-1541) is...
"Credited with discovering Parseltongue"
Obviously this conflicts heavily with later canon mentions of Salazar Slytherin and Herpo the Foul but it is worth noting that JKR did write the cards seen in the EA games, even if the cards themselves aren't strictly considered canonical:
Q: Did you actually write the information that ended up on the Famous Wizard cards? For that matter, what about the spells in the films? Did you invent those or did Steve Kloves? And why were new incantations created for the movie in the first place? (Example: "Incendio" to "Lacarnum Inflamari"?
JKR: Yes, I wrote the information on the original Famous Wizard cards. As you have notices, a few of them have now popped up on the 'Wizard of the Month' cards on my website desk.