I believe this is somewhere in Hirsch's book on differential topology, but I am not sure. The question is:

When is $S^m \times S^n$ parallelizable?

If $m$ and $n$ are even, $\chi(S^m \times S^n) = 4$, hence the Euler class is nonzero and the manifold cannot be paralleizable.

If, however, at least one of $m,n$ is odd then one can use that spheres have stably trivial tangent bundles and 'borrow' a line bundle from the other sphere...

I believe this is somewhere in Hirsch's book on differential topology, but I am not sure. The question is:

When is $S^m \times S^n$ parallelizable?

If $m$ and $n$ are even, $\chi(S^m \times S^n) = 4$, hence the Euler class is nonzero and the manifold cannot be parallelizable.

If, however, at least one of $m,n$ is odd then one can use that spheres have stably trivial tangent bundles and 'borrow' a line bundle from the other sphere...