It's not so much a reversal of policy as a false perception. The LGBT movement is at its own stage in Russia, but has been used by western backed agencies to stir up controversy. The issue has been used to portray Russia as authoritarian, but this shows the reality in the streets. Their participation in the parade did not break the law, but the anti-gay mob threatened to take the law into their own hands.
IOM, the only reason mainstream westerners buy into this type of demonization is to help them avoid looking at things back home.
Authoritarian is as authoritarian does, and if this was going down in NYPD's beat, the feds would have wire tapped the anti-gay side, befriended them, trained and funded them, then entrapped one of them into planting a bomb in the crowd.
So yeah. This is more like the true face of Russia, and not some sudden policy change.