So, once upon a time, Discord was causing a heckin big kerfuffle all over Equestria, really fucking everybody's shit up with his chaos magic. And there was this one griffin named Auric, who'd had enough of Discord's bullshit, and really let him have it verbally. Well Discord, thankfully, has never really been one for killing creatures, so he turned Auric into his immortal servant sidekick. He made him his bitch. I don't remember the exact details; it's been a while. But the point is, Discord then got stoned himself by those pesky elements of harmony, leaving Auric an immortal catbird with nothing to do. He got all angsty and shit, but eventually found purpose through Seriously, go read "The First Time You See Her". It's far better than this dreck..
Right so, there's this immortal gryphon. And when you're immortal, you try a lotta freaky shit, like flagpole sitting and bloodplay. But that wasn't Auric's scene. No, he had other ideas. Like making sweet sweet half-bird love to cockatrices.
Auric fucked a cockatrice.
Now, I'm going to go ahead and assume it was a sapient cockatrice that was able to give consent. Jack Harkness test, and all that. Don't fuck non-sapient creatures, kids. That's a big no-no.
So yeah, this half-lion-half-eagle did the horizontal somba with a half-dragon-half-chicken. And you know what? The cockatrice got pregnant.
Now, this was pretty soon after Discord's reign, so there was still a lot of residual wonky magic gunking up everybody's biz. So the cockatrice got pregnant. And laid an egg.
And as it turns out, the bird parts of these creatures were the parts that genetically were happening. Or something. So when that egg hatched, a beautiful baby cockatiel emerged. From his mother, he got the "cock" in his name and the ability to turn ponies (and cats) to stone by making eye contact with them. From his Father, he got immortality. From both, he got birdness. So there you have it. That's why there's a cockatiel in this story that can turn ponies to stone. And is immortal apparently.