TL;DW:

Given a video like:
youtube.com/watch?v=VideoID&list=PlaylistID

Change it to:
youtube.com/watch/v/VideoID?start=188&end=193

Where start and end are the number of seconds to start and stop playing the clip. In this case, we want to start at 3 minutes and 8 seconds, so 3*60+8=188, which becomes our start time.

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Example:
youtube.com/watch?v=gcucs4VBy4c&list=PLqzpS5Mv3UKpE8GutmFn13gsExFSeTScr

Becomes:
youtube.com/v/gcucs4VBy4c?start=188&end=193

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