I think ever since the Soviet Union fell, Russia has tried to restore it's Orthodoxy and pre-revolutionary culture. While it's a good thing for them to have certain freedoms, they probably feel threatened by Western philosophies trying to enter the country. This is why it's hard for Protestant and Catholic missionaries there. I certainly disagree with Russian's religious restrictions on Protestants and Catholics, but really, does the Russian government really think any Orthodox Christian will fall for a 45 minute Bible study, sitting down, and only twice a week, to the Divine Liturgy?
The same effect is on Homosexuals. While much of the West has tried to turn the gay pride movement into some sort of 1964 civil rights movement, Eastern European culture is very different. Notice that almost all supporters of this gay pride march in Moscow were people from America, Canada, and Western Europe! And only persecuting this movement will turn it into a crusade and Westerners will make a big deal of it.
The advantage of allowing these parades is that it might prevent it from getting bigger. If the event isn't illegal, than the parade will be boring to go to in the first place. The disadvantage of allowing the parade is that Moscow citizens may attack it and further it's media attention and Western sympathy. Maybe the mayor could just say it's not allowed do to the safety of the protesters.
Showing it hostility will only put a bad name on Orthodox Christians, especially for what little Westerners know of them.