In Harry Potter, it says that Dumbledore can speak Parseltongue. Where is this proved, and where does he learn it?
No, in an interview, J.K. Rowling confirms that Dumbledore understands Parseltongue (Parseltongue is the language; a Parselmouth is someone who speaks Parseltongue). There is no canon confirmation that Dumbledore is a Parselmouth or that he speaks Parseltongue. We only know that he understands it. He also understands Mermish and Gobbledygook (Goblin language), as well as English.
There is discussion about Dumbledore and Parseltongue at Why didn't Dumbledore hear the Basilisk?
Delailah: How does Dumbledore understand Parseltongue?
J.K. Rowling: Dumbledore understood Mermish, Gobbledegook and Parseltongue. The man was brilliant.
Interview with J.K. Rowling - 7.30.07 - The Leaky Cauldron
Ah...no. He didn't. He understood Parseltongue.
Imagine, for a moment, a magical creature that spoke a language so ridiculous that the human tongue could not wrap itself around the syllables. You could listen, and have the creature translate, and remember that 'buflihysnovbrywesadfvbnwhiodsf' means 'toilet seat', but you couldn't actually pronounce 'buflihysnovbrywesadfvbnwhiodsf', now could you? This is the situation with Parseltongue.
Dumbledore knows what the various sounds mean, probably by hearing them over and over, but he can't hope to be able to speak a word.