I'm sorry, but I've searched the forum multiple times and cannot find information on this question.
Did the individuals who participated in the Moscow Victory Parades for the 60th anniversary in 2005 or the 65th anniversary in 2010 receive a special medal from the Russian Government? I am not referring to the standard anniversary medal. I've heard something about a medal which illustrates the moment from the original Victory Parade in 1945 when the captured flags of the Third Recih were displayed.
Did various ministries, below the national government level, award medals for participation in the parades?
There are no federal Medals for participating in the victory parade. There are 2 ministerial medals (Defence and Emercom) for participating in victory parades and tens if not hundreds of "organisational" awards for participating in parades in most of the major cities of Russia.
I didn't hear of State medals being awarded for the actual parade, but Ctahhr could be correct that some ministerial medals were given. As you've pointed out, the jubilee medals were given to participants in commemoration of the anniversary and not solely for parade participation.
Here is a cased example. These were awarded for participation in the 65th, 66th, 67th, 68th, 69th Victory Day parades and will probably be used for the 71st and onward as well. It was not used for the 70th.
Here is an unissued document, produced by the "Awards Factory". Modern RF awards are subject to contract and production quality fluctuates. I have my favourite mints and this one comes about half way down the list when it comes to quality.