See the following for a list of all the possible parameters:
https://developers.google.com/youtube/player_parameters
In the theme, copy/paste the iframe embed code from the YouTube share panel into the Embed Code field. See this article for a detailed walkthrough.
Add your desired URL parameters to the end of the URL in the embed code. Example:
<iframe id="ytplayer" type="text/html" width="640" height="390"
src="http://www.youtube.com/embed/M7lc1UVf-VE?autoplay=1&rel=0"
frameborder="0"/>
In the above example, a ? is placed after the URL, followed by two parameters: autoplay = 1 and rel=0. This sets the video to autoplay on load, and not show related videos at the end.
This article covers how to do this to force transparency on the iframe if your videos are overlapping the navigation drop-downs or other elements on the page due to Flash.