We show that, on an oriented Riemannian 4-manifold, existence of a non-zero parallel spinor with respect to a spinc^cstructure implies that the underlying smooth manifold admits a Kähler structure. A similar but weaker condition is obtained for the 4-manifold to admit a symplectic structure. We also show that thespincspin^cstructure in which the non-zero parallel spinor lives is equivalent to the canonical spinc^cstructure associated to the Kähler structure.