If that doesn't happen then Picard, Sisko & Co as we know them has quite simply been earsed from existance. It's not a parallel universe situation at all... .
Why would Picard and Sisko be erased? Earth didn't blow up. There's no reason to think that Picard's ancestors living in a wine growing area of France wouldn't meet and marry the same people they did in old timeline. Same for Sisko's ancestors.. and certainly the prophets could still have chosen Sisko's father as the mate of one of them.
As Star Trek has shown, parallel universes have nothing in common with other universes they're parallel to. Just look at how different the mirror universe is, for example, with everyone still marrying the same people and having the same offspring with the same names who all happen to have the same jobs on the same starship... oh, wait...
DOH!
