Parseltongue is the ability to speak to snakes. Are there other types of tongues besides parseltongue that you can use to talk to other types of animals? Maybe doggietongue or kittytongue?
Yes. Kinda. Maybe. Canon isn't very clear on this.
Wormtail was able to communicate to rats (Voldemort discusses that in GoF at the end, explaining how Wormtail found him in Albania). Whether it was a special language, or animagus "mind-meld" isn't clarified.
And Padfoot/Sirius could communicate with Crookshanks in PoA. Somehow. Canon's murky on how.
There's reason to believe that Dumbledore was able to communicate to Fawkes, but that seems to be non-verbal so shouldn't count.
Magical creatures that are sentient of course have their own languages (Goblins, Mermen, Trolls, Giants)
The answer is yes.
Dumbledore was able to speak mermish, and it was allegedly said that Barty Crouch Sr. could speak it. This is a quote found on the Harry Potter Wiki,
"Albus Dumbledore was a known non-Merperson speaker, and Percy Weasley claimed that Barty Crouch Sr. could also speak it."
The quote can be found here, under History, second line.
The more obvious answer are other human languages. Bulgarian for example, in 1994, at the Quidditch World Cup, between Oblansk, the Bulgarian Minister, and Cornelius.
Other tongues include Gobbledegook, the native language of the goblins. Harry Potter Wiki states: "Barty Crouch Sr., Dirk Cresswell, and Albus Dumbledore are known non-Goblin speakers."
Quote found here, under Gobbledegook, first paragraph, second line.
Runes can also be a possible other language, and a more far fetched tongue can be Troll.