'. Let Cg be a smooth projective irreducible curve over C of genus g ≥ 1 and let {p 1 ,..., p s } be a set of distinct points on Cg. We fix a nonnegative integer l and denote by M g (p, λ) the moduli space of parabolic semistable vector bundles of rank r on Cg with trivial determinant and fixed parabolic structure of type A = (λ 1 ,..., λ s ) at p = (p 1 ,..., p s ), where each weight λ i is in P l (SL(r)). On M g (p, λ) there is a canonical line bundle L(λ, l), whose global sections are called generalized parabolic SL(r)-theta functions of order l. In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.