I noticed this while watching WWII in color on Netflix. I saw British, German, and I think Belgian armies doing this. This video includes examples of what I mean around 0:35 - https://www.youtube.com/watch?v=4BAoYNV5O78 (I sincerely couldn't find a version of this without the wii music, but it's kind of funny to watch anyway)
In the United States, at least, it's turning to the reviewing officers or guests of honor. The leader of the group with also salute.
They are currently marching past the reviewing officers and guests of honor as /u/hallcha stated. Between 0:30 and 0:35 you'll notice the camera is on the dignitaries, this is the people who are getting the eyes right. This constitutes the formed body saluting the reviewing officers.
During most of the parade they will march with their heads forward. This changes when they pass the reviewing officers who need to be saluted. I'm unsure about other armies but in Australia the person leading that group of soldiers will use the command "Eyes - Right" and on the sound of 'right' the soldiers will snap their heads, except for the soldiers in the right hand file (notice how they look straight?).
If you pause at the 0:56 mark you'll be able to see some officers in front of their formed troops and the officers are saluting.