Túrin spoke the Mannish tongue of the House of Hador and also Sindarin (in a noticeably archaic Doriathrim way). On the other hand the petty-dwarves being largely isolationist and hating the Eldar, would be unlikely to know Sindarin, and it is also hard for me to imagine that they would speak Mannish tongues well.
Yet Túrin (and even his men) had no linguistic hardship when communicating with the petty-dwarves; in what language was this?
Is there really any other possibility, possibly Túrin, owing to the Khuzdul influences on his mother-tongue (that the Hadorians are said to have acquired before passing to Beleriand) and perhaps by conversing with Dwarven craftsmen visiting Menegroth, knew enough Khuzdul to somehow understand Mîm? Or did the petty-dwarves profess knowledge of Sindarin or some Edain pidgin?