Russian site states that kid's voice ---- Aleksei's one can be heard between 39th and 41nd second of that recording, as well as Eduard Vladimirovich's and Nicholas'.
http://www.youtube.com/watch?v=HYZ3IuHoLJk&feature=related
EDIT: This is speculation, I don't know what to think actually. Others?
This is not speculation. The probability that the voice of the Tsarevich Alexei sounds in the video is very big.
Firstly it is certainly the voice of children. I have never heard "out of breath and voice" ( as Dust_of_History writes ) on the military men's parade, even more to leave it on record.
I see no reason why the "bad" voice have left on the recording close to the Emperor' voice, unless it is the voice of the Crown Prince.
Secondly, other children in the video and photos on the Russian parade does not occur.
And thirdly, Emperor Nicholas II often took Alexis with him to military parades . Beginning in 1915, when the Emperor took over the Supreme Commander during the war, almost always with him was the Crown Prince.
As, for example, in this clip:
http://youtu.be/wMs2h-NBTvw