Stably parallelizable iff framable in large codimension

Statement

Suppose $M$ is a differential manifold. Then, the following are equivalent:

• $M$ is a stably parallelizable manifold i.e. its tangent bundle is a stably trivial vector bundle
• There exists a manifold $N$ such that $M$ can be embedded as a framable submanifold of $N$
• There exists a manifold $N$ of dimension at most $2dim(M)+1$, such that $M$ can be embedded as a framable submanifold of $N$