Well, the 1-sphere and the 3-sphere are both Lie groups, so they are easily parallelized by taking a basis at the identity and moving that around by actions of the corresponding Lie group.
The 7-sphere isn't a Lie group, but the octonionic structure is what allows for the parallelizability, just as the complex and quaternionic structures on the 1-sphere and the 3-sphere respectively allow them to be Lie groups and thus parallelizable. I'm not quite sure how this works, though, since I can't think of how the parallelization of the 7-sphere wouldn't lead to a Lie group structure of some sort.
The parallelization of an n-sphere conversely gives rise to a division algebra, which would explain why there is the restriction to the 1-, 3-, and 7-spheres.
More details I don't quite know.
There's also this:
complicated algebraic geometry/topology proof for why S(4k-1) is not parallelizable for s > 2 I haven't quite gotten through all of it and it requires a lot of reference-checking.
The people to go to for proof are R. Bott, J. Milnor and M. Kervaire, the last of whom is the author of the link I provided above.
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