Actually, that's the many-worlds hypothesis of quantum theory. Parallel universes refers to something else.
> This proliferation of multiple futures from any moment would tend to provide a way for doomsdays to be circumvented.
Not exactly circumvented. In the many-worlds picture, all outcomes occur, including the one predicted by the Doomsday argument.
> If you are cloned into N futures in this moment, and you're killed in N-1 of them, then you don't know it ...
True but not compelling, because quantum theories don't require each of the many worlds to be occupied by a conscious observer. Were this not so, the universe couldn't have evolved to the point where consciousness was first possible.
As far as I know, that's a requirement for any thought experiment involving SSA vs. SAA.
For starters, we have no idea of how likely our universe is; we know what numerator is 1, but what goes on the denominator? Is there even such a number? If so it is something so huge that the probability is vanishingly close to zero.
Then what about the parallel universes hypotheses: what if the universe is constantly fragmenting into multiple futures. In one future, I will submit this comment and close the tab, then check my e-mail. In another, I keep the tab open and go make a coffee.
This proliferation of multiple futures from any moment would tend to provide a way for doomsdays to be circumvented.
If you are cloned into N futures in this moment, and you're killed in N-1 of them, then you don't know it; by the anthropic principle itself, only the surviving future matters; there is no consciousness in the others. The suriviving future has no idea about the size of N, either.