To make a video you've embedded on your site start where you want it to, you have to add a time-to-start command to the embed code you get from youtube.
It tells the video to start so many seconds into the video. So watch the video. See how many seconds in you want it to start.
Then, add &start= to the embed code followed by the number of seconds.
It has to be done twice as in the example below:
<object width="425" height="344">
<param name="allowFullScreen" value="true"></param>
allowfullscreen="true" width="425" height="344">
If you don't want to embed the video but only link to it, you have to put the time-to-start instructions right onto the URL you are linking to.
First you add the code, #t=, then the number of minutes followed by the letter m. Then the number of seconds followed by the letter s.
For instance: #t=3m20s.
Here's an example of the full URL: