On Pottermore it states that Parseltongue is rare except for in the bloodline of Salazar Slytherin. Why is Parseltongue common in Slytherin's bloodline, but nowhere else?
Salazar was a Parselmouth, and his descendants inherited it.
The natural ability to speak Parseltongue is an inherited trait. Salazar Slytherin was a Parselmouth, and his descendants tended to inherit it as their ancestor had it.
“You can speak Parseltongue, Harry,’ said Dumbledore calmly, ‘because Lord Voldemort – who is the last remaining descendant of Salazar Slytherin – can speak Parseltongue.”
- Harry Potter and the Chamber of Secrets, Chapter 18 (Dobby’s Reward)
J.K. Rowling says in an interview that the ability is passed down through the Slytherin bloodline.
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. However Ron was with Harry when he said one word in Parseltongue, which I do not know so I cannot duplicate for you, but he heard him say "Open," and he was able to reproduce the sound. So it was one word. Whether he could learn to speak to snakes properly is a separate issue. I don't think he could. But he knew enough, he was smart enough, to duplicate one necessary sound.
- J.K. Rowling at Carnegie Hall (October 20, 2007)
It’s common in the Slytherin bloodline because they’re more likely to inherit it. However, it’s not stated that there are no Parselmouths not of Slytherin’s bloodline, and the pure-blood families are all interrelated anyhow, so it’s likely that many more wizards would have a chance to inherit the natural ability to speak Parseltongue than the idea of it being tied to the Slytherin bloodline implies.
We don't know why parselmouths are rare outside of Slytherin's bloodline
As Bellatrix said, being a parselmouth is something that is inherited from one's ancestors, which explains why it shows up for many individuals in Slytherin's bloodline.
I don't remember if we are ever given information about how likely it is for a descendant of Slytherin to be a parselmouth. The last of the Gaunts, Marvolo and his children Morfin and Merope (as well as Merope's descendants Voldemort and Delphini), were all parselmouths, but we are told that the Gaunt family often practiced inbreeding, which it could be conjectured might have increased the prevalence of parselmouths.
There are some pieces of evidence that might indicate that not all of Slytherin's descendants possess this ability:
the Pottermore page on "Ilvermorny School of Witchcraft and Wizardry" says that Isolt Sayre, a descendant of Salazar Slytherin, "could not speak the language [Parseltongue]" (although the context for this quote makes it not as clear as I would like, so I have asked a separate question: Was Isolt Sayre a Parselmouth or not?)
The memory of Tom Riddle tells Harry that they are “Probably the only two parselmouths to come to Hogwarts since the great Slytherin himself,” even though Gaunts had previously attended the school (e.g. Corvinus Gaunt): see the answer here. But he might just have been wrong, as alexwlchan suggests: we know that Voldemort liked to think of himself as unique, and a result had a tendency to underestimate other people (for example, he may have assumed that nobody would find the Horcrux that he hid in the Room of Hidden Things).
There seems to have been at least one Slytherin descendant who was surely not a parselmouth because she was not magical at all: Isolt Sayre's daughter Martha, who was a squib.