I recently wanted to embed a YouTube video in a page, but wanted the video to start at a specific time in the middle. There are a lot of obsolete instructions out there based on the way YouTube embedding worked in the past. I finally figured out that the trick is to use the code that YouTube gives you under the “Embed” option, but add a ?start=XXXX parameter at the end of the URL, where XXXX is the start time you want to show in seconds.
Looks like this:
<iframe width="640" height="480" frameborder="0" style="width: 640px; display: block; margin-left: auto; margin-right: auto;" src="https://www.youtube.com/embed/TEbquFE8AN8?start=3082" allowfullscreen="">