A parallel adaptive mesh refinement algorithm has been incorporated into the side-centered finite volume method [Sahin, A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations. J. Non-Newtonian Fluid Mech., 166 (2011) 779-791] in order to obtain highly accurate numerical results for viscoelastic fluid flow problems. The present recursive mesh refinement algorithm is based on a conformal refinement of unstructured quadrilateral/hexahedral elements with templates based on 1:3 refinement of edges. In order to transfer cell-centered data between source and target meshes, a second-order conservative interpolation (remapping) technique similar to the work of Menon and Schmidt [Supermesh construction for conservative interpolation on unstructured meshes: An extension to cell-centered finite-volume variables. Comput. Methods Appl. Mech. Eng., 200 (2011), 2797-2804] are employed and the approach has been extended for side-centered data. The proposed framework has been applied to the classical benchmark problem of an Oldroyd-B fluid past a confined circular cylinder in a rectangular channel and a sphere falling in a circular tube. The calculations confirm that high accuracy can be achieved with the present adaptive mesh refinement. (C) 2016 Elsevier B.V. All rights reserved.