The phrase "that's my bark to peel" refers to an artistic concept wherein an artist is to decide the meaning and content of their art. It can also be used in a similar way to "that's my axe to grind", meaning that it's your business to deal with.
"That's my bark to peel" as in "that's my art to create", "that's my bark to peel" as in "I dictate the meaning of my piece". In some ways this idiom can lie directly contrary to the artistic concept of The Death of the Author, a concept which has the viewer as the decider on the pieces meaning.
This idiom was created by Canadian writers Andy Zuliani and Cassie Mifflin based off Mifflin's short story "Cold Pudding and the Last Great Maritimer" wherein she included a line after her bark was stolen by artist Ruben Moller where she said "That's my bark to peel". Andy would then go on to coin the phrase's meaning while discussing her work and then further what that moment meant, telling her that "If a professor is ever making you tear apart your work more than you'd like, just tell them 'that's my bark to peel.'"
Andy: "If a professor is ever making you tear apart your work more than you'd like, just tell them 'that's my bark to peel.'"