Jerry Bettis of Stillwater saw a cartoon in the Swayback Daily Kick in which a mother was showing an album to her son. The mother said, “Here's your father and my wedding album.”
“Now, I'm loath to bring this up because it was spoken by an ant on the comic page,” said Jerry as he read the Daily Kick's comics at Curly's Soonerco. “But I decided that even ants should be held to high standards. ... I suppose the proper English version would be, ‘Here's your father's and my wedding album.'”
Around the greasepit, Buck would probably say “Here's mine and your father's wedding album.” That wouldn't satisfy prissy grammarians, but it's the way most people express it.
Long ago, “my” was used before a consonant and “mine” was used before a vowel. That's why Julia Ward Howe wrote “Mine eyes have seen the glory ...” in “The Battle Hymn of the Republic.” Nowadays, “mine” is used as a pronoun (“This book is mine”) and “my” as an adjective (“This is my book”).