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
I always assumed he learned it from the only other confirmed parselmouth in the series; Tom Riddle– ValorumJun 18, 2014 at 8:47
6@Richard -- Just ... no. Tom Riddle avoided Dumbledore like the plague because Riddle showed his psychopathic hand to Dumbledore at his excitement at learning he was a wizard. No way was Tom Riddle ever going to tip his hand to Dumbledore again. Dumbledore knew Tom was a Parselmouth, but I can't imagine under any circumstances Tom Riddle giving Albus Dumbledore Parseltongue lessons and canon supports Tom Riddle's secretive nature, especially toward Dumbledore. Now, Slughorn? Him, I could see Riddle giving Parseltongue lessons to! :) Jun 19, 2014 at 1:17
He must have spent at least a modicum of time with him– ValorumJun 19, 2014 at 4:37
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.
5But Ron could speak a word of Parseltongue ...– Rand al'Thor ♦Apr 18, 2017 at 11:33