iOS 7 seems to have removed the ability to view individual audiobook chapters in the music app. It has also replaced the chapter length with the length of the entire book in the time scrubber. When tapping the top right-hand corner to access the list of chapters, now it only shows one file for the whole book, whereas before it listed all the chapters, just like it lists all the songs in an album.
The earliest reports came in over a week ago, and over a hundred different people have confirmed the initial report. One even added an updated yesterday which informs us that iOS7.0.2 does not fix the bug.
Needless to say this is a serious issue. It's not unusual for an audiobook to be 8 or 10 hours long, or even longer. That is simply too long to search for when trying to find a specific location, so listeners have been using the chapter divisions in order to simplfy navigation.
There's been no word from Apple on when this issue will be fixed, but I do know that Apple is planning to release more updates to iOS7 in the near future. Until then, you might want to try switching back to one of these:
Snarky comments aside, I have also been told that the Audible app for iPad and iPhone doesn't have this feature either so that option is out.
Thanks @mdash for the tip!