Isn't the 2-dimensional sphere a counterexample? If $f$ has degree 1, then it's homotopic to the identity, so $f^*(TM)\cong TM$. If $f$ has degree $-1$, then it's homotopic to the antipode map $a$, so $f^*(TM)\cong a^*(TM)\cong TM$, where the last $\cong$ is evident if we embed the sphere in the standard way in $\mathbb R^3$ so that the tangent spaces at a point and its antipode are the same as vector spaces. Since a diffeomorphism must have degree $\pm1$, we have $f^*(TM)\cong TM$ in all cases. But even-dimensional spheres don't admit even one nowhere-vanishing vector field, because they have non-zero Euler characteristic, so they are far from parallelizable.
