You can add a start parameter to the embed code that YouTube generates for you.
<iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/Iq0XJCJ1Srw?rel=0&start=56" width="560"></iframe>
Updated Apr 2013
https://yousufrafi.wordpress.com/2012/08/24/start-a-youtube-video-at-a-certain-timestamp/
From 2011
From https://www.mattcutts.com/blog/start-youtube-video-minutes-seconds/
https://www.youtube.com/watch?v=8tMuSVVljg8#t=17m08s
The “#t=17m08s” takes you to 17 minutes and 8 seconds in a video.
Embedded code.
<object height="385" width="640"><param name="movie" value="https://www.youtube.com/v/8tMuSVVljg8&hl=en_US&start=1028"></param><param name="allowscriptaccess" value="always"></param><embed src="https://www.youtube.com/v/8tMuSVVljg8&hl=en_US&start=1028" type="application/x-shockwave-flash" allowscriptaccess="always" width="640" height="385"></embed></object>See here for the example working.
This guide gives some other good pointers, "The default (and recommended) embed size is 480x385 for a 4:3 video and 640x385 for 16:9 content."
Here is an example of using 640x385 for 4:3 video.
