Okay, so I found the result that the tangent-bundle of any product of spheres is parallizable, given that some element of the product is either $S^1$, $S^3$, or $S^7$. I prove this as follows, first noting that since the higher-spheres are simply-connected, they $\Phi^{\mathbb{R}}_1(S^i)=1$, since rank-1 vectorbundles are in correspondence with double-covers (over a paracompact space). Thus for $j\in \{1, 3, 7\}$ have (omitting the pull-back map of the projection which occurs on most terms, and switching $1$ with the normal bundle by above) that $T(S^i\times S^k)=T(S^i)\oplus T(S^k)$$=i\oplus T(S^k)=$$(i-1)\oplus N(S^k)\oplus T(S^k)=(i-1)\oplus (k+1)$$=k+i$ if $k>1$ and if $k=1$ it is trivial, now factoring and using the trivial bundles to parallize the other spheres in this way, we get the conclusion.
My question is does this extend? I want to say that it is true for all products, due to the hope that since the tangent-bundle is associated to the clutching-map using the upper and lower hemispheres of the rotation matrices, that the product will be the matrix-direct-sum of the rotation matrices which I want to say is null-homotopic, but I don't know enough to try to go deeper into this aurugment.
Does anyone have a solution to this problem?