Hi, thank you for contacting JustAnswer.com. My name is Russell.
Andrew is unfortunately not responding at the moment. I hope I can help. I will do my best to provide the right answer to your question.
A playlist (that isn't your own playlist - and I know the difference since I have numerous playlists on my YouTube account - ) necessarily begins at the beginning, or at the first video listed.
When you interrupt playlist playing, you will probably have to begin it over again - unless you simply pause the video, and then un-pause it later, keeping it in a Tab of your browser until then.
What browser are you using? on what platform or what computer type?
A YouTube playlist ought to come up with a sidebar to the right, listing each video in the list, the earlier ones on the top of the list, the last video in the playlist at the bottom.
There are YouTube settings that allow you (if logged in to your YouTube account) to play through a playlist differently. They may be a factor in what you're asking about, too. A setting can either be (if I recollect correct) Shuffle (or random order), play-through (more or less), or neither. If you're set to 'Neither', playing one video from in the middle of the playlist will simply play that video and leave off; you would need the setting for play-through more or less. Which in my browser, at least, is called Play All (actually not a setting. But it plays all the playlisted videos through in order. The two settings are either Repeat Playlist, or Shuffle... and they are represented by little icons with two arrows each, which are found just beyond but near the upper right of the video being played.)