I think there is somewhat a distinction between being a Parselmouth and understanding Parseltongue.
Let me explain: Harry was a Parselmouth. He had a natural ability (albeit - given to him by Voldemort) to speak to snakes and understand them. Therefore, he could speak Parseltongue without realizing it.
On the other hand, it must be possible (although surely very difficult) to learn the language as well. For example:
Spoiler:
Ron manages to open the chamber of secrets in DH by hissing.
I think Dumbledore learned Parseltongue (and therefore -- understands it and speaks it as well), just as he learned Mermish. But he was not a (natural) Parselmouth, only Slytherin, Voldemort, the Gaunts and Harry are known to posess this special, natural ability.
(This can be compared loosely to the idea of a Metamorphmagus. Anyone can transfigure himself and thus change his appearance, but there are those who are born with a natural ability to do so easily. I think Parselmouths -- as opposed to understanding Parseltongue -- is the same idea).
As to the argument that Dumbledore didn't hear the basilisk, there could be several answers to this: first of all, who says he didn't hear it? It's also possible that he heard it, but not being a natural Parselmouth he simply didn't realize it was Parseltongue. Also, it's possible that Dumbledore was simply never in the area when the basilisk went by...
Well maybe he can understand it, but I don't think he can speak it.
The only way that you could get the ability to speak parselmouth is by inheritance (if you father or mother is one). And since JK said Harry wasn't related to Dumbledore...
i agree with this idea i think u have to be slytherins descendant to be able to speak but i dont think you have to be his descendant to be able to understand it he might have been able to learn
Wonder how Dumbledore learned it?
Was Gellert Grindenwald a parseltounge? Because he might have ben taught by him. _________________ Let me baby dragon hatch!