Are there any simply connected parallelizable 4-manifolds?

A simply connected smooth compact 4-manifold has its homology concentrated in even degrees, by Poincare duality. Thus its Euler characteristic is positive. By the Poincare-Hopf index theorem, such a manifold can have no nowhere vanishing vector field, and so is certainly not parallelizable.

At least $S^3$ is a lie group so parallelizable.