The tangent bundle to a smooth structure on $S^7$ is classified by a map $S^7 \to G_7(R^{\infty})$. By the [exact sequence for a fibration][1] for the fiber bundle $O(7)\to V_7(R^\infty)\to G_7(R^\infty)$, we see that $\pi_7(G_7(R^\infty)) = \pi_6(O(7))$. But $\pi_6(O(7))=0$ (I found a table A1.1.3.2 of homotopy groups of orthogonal groups [here(pdf)][2], since this isn't in the stable range of Bott periodicity), so the tangent bundle is trivial, i.e. parallelizable. 


  [1]: http://en.wikipedia.org/wiki/Homotopy_group
  [2]: http://felix.physics.sunysb.edu/~abanov/Teaching/Spring2009/Notes/abanov-cpA1-upload.pdf