Though computational techniques for two-dimensional viscoelastic free surface flows are well developed, three-dimensional flows continue to present significant computational challenges. Fully coupled free surface flow models lead to nonlinear systems whose steady states can be found via Newton's method. Each Newton iteration requires the solution of a large, sparse linear system, for which memory and computational demands suggest the application of an iterative method, rather than the sparse direct methods widely used for two dimensional simulations. The Jacobian matrix of this system is often ill-conditioned, resulting in unacceptably slow convergence of the linear solver; hence preconditioning is essential. In this paper we propose a variant sparse approximate inverse preconditioner for the Jacobian matrix that allows for the solution of problems involving more than a million degrees of freedom in challenging parameter regimes. Construction of this preconditioner requires the solution of small least squares problems that can be simply parallelized on a distributed memory machine. The performance and scalability of this preconditioner with the GMRES solver are investigated for two- and three-dimensional free surface flows on both structured and unstructured meshes in the presence and absence of viscoelasticity. The results suggest that this preconditioner is an extremely promising candidate for solving large-scale steady viscoelastic flows with free surfaces.