Did you purchase the audiobook in the iTunes/iBooks/Music app?
If so, the good news is you don't need to copy the audiobook from one iPad to the other. Almost all content from the iStores can be downloaded again for free on any device that uses the same iTunes account.
Your new iPad may be running iOS 8.4, in which case the audiobooks have moved to the iBooks app (they used to be in the Music app).
If the audiobooks are still in Music (pre iOS 8.4) then go to the Purchased tab in Music. You should see your audiobook listed, and can tap on it to download again. If in iBooks, the same trick should work. To see an audio book in iBooks you must have All Books or Audiobooks selected as a category (top center).
Hint: If you use the Not On this iPad tab at the top of the Purchased view you can limit what you are looking for to only the items you might want to download.
If the book was an Audible.com book, you'll need to download and sign into the Audible.com site. Like iTunes, your old content should be free to download again. I can't offer details, since I don't have this app.
If you got the audiobook by some other method, then please describe where and how you purchased it, and got it onto the iPad to begin with. With luck someone will have dealt with that method before.