It used to be easy- there was an option in embed to “Start video at”
and it was easy to custom pick sizes using the old embed code.
I wanted to start a video at 39:27 in- so I had to multiply 39x 60 + 27 to come up with 2367 seconds
Copy the standard embed code: (I’ve removed starting and ending brackets)

iframe width=”640″ height=”360″ src=”//www.youtube.com/embed/6U5z19pvCbA” frameborder=”0″ allowfullscreen></iframe

Then put this on the end of the video url “?start=2367?:

iframe width=”640″ height=”360″ src=”//www.youtube.com/embed/6U5z19pvCbA?start=2367” frameborder=”0″ allowfullscreen></iframe

Voila- it starts at the right time.