Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra ?2 (or ?2d more generally) are presented by using recursive constructions. As a typical example, an embedding of CAR onto the U(1)-invariant subalgebra of ?2 is constructed explicitly. Generalizing this construction to the case of ?2p, an embedding of CAR onto the U(1)-invariant subalgebra of ?2p is obtained. Restricting a permutation representation of the Cuntz algebra, we obtain the Fock representation of CAR. We apply the results to embed the algebra of parafermions of order p into ?2p according to Green's ansatz.