I tried the method of @mystic11 ( https://stackoverflow.com/a/11422551/506073 ) and got redirected around. Here is a working example URL:

http://youtube.googleapis.com/v/WA8sLsM3McU?start=15&end=20&version=3

If the version=3 parameter is omitted, the video starts at the correct place but runs all the way to the end. From the documentation for the end parameter I am guessing version=3 asks for the AS3 player to be used. See:

end (supported players: AS3, HTML5)

Additional Experiments

Autoplay

Autoplay of the clipped video portion works:

http://youtube.googleapis.com/v/WA8sLsM3McU?start=15&end=20&version=3&autoplay=1

Looping

Adding looping as per the documentation unfortunately starts the second and subsequent iterations at the beginning of the video: http://youtube.googleapis.com/v/WA8sLsM3McU?start=15&end=20&version=3&loop=1&playlist=WA8sLsM3McU

To do this properly, you probably need to set enablejsapi=1 and use the javascript API.

FYI, the above video looped: http://www.infinitelooper.com/?v=WA8sLsM3McU&p=n#/15;19

Remove Branding and Related Videos

To get rid of the Youtube logo and the list of videos to click on to at the end of playing the video you want to watch, add these ( &modestBranding=1&rel=0 ) parameters:

http://youtube.googleapis.com/v/WA8sLsM3McU?start=15&end=20&version=3&autoplay=1&modestBranding=1&rel=0

Remove the uploader info with showinfo=0 :

http://youtube.googleapis.com/v/WA8sLsM3McU?start=15&end=20&version=3&autoplay=1&modestBranding=1&rel=0&showinfo=0

This eliminates the thin strip with video title, up and down thumbs, and info icon at the top of the video. The final version produced is fairly clean and doesn't have the downside of giving your viewers an exit into unproductive clicking around Youtube at the end of watching the video portion that you wanted them to see.