We present a proof of a theorem by Dold and Whitney, according to which a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our proof is based on an argument by R. Kirby on the classification of SO(4)-principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.
