You can use the YouTube class that is available in the MyToolkit library for Windows Phone, take a look:
Example to play a YouTube video using a MediaElement, but using the GetVideoUriAsync method: XAML with MediaElement:
XAML with MediaElement:
<MediaElement x:Name="player" Width="480" Height="320" />
The code to start the video in the player (for example, on a clicked button):
var url = await YouTube.GetVideoUriAsync(youTubeId, YouTubeQuality.Quality720P);
if (url != null)
{
player.Source = url.Uri;
player.Play();
}
else
// TODO show error (video uri não existe)