From what I can tell it is an inherited skill from wizard to wizard. According to this question: Was there a Parselmouth before Salazar Slytherin? Herpo the Foul seems to be the first or one of the first who were able to speak Parseltongue.
And from what I have researched, he was an Ancient Greek wizard and there is nothing of significance if he was related to Salazar Slytherin.
So my question is, how does Salazar Slytherin speak parseltongue? (which could possibly be answered by if he was actually related to Herpo the Foul)