An essential ingredient for the discretization and numerical solution of coupled multiphysics or multiscale problems is stable and efficient techniques for the transfer of discrete fields between nonmatching volume or surface meshes. Here, we present and investigate a new and completely parallel approach. It allows for the transfer of discrete fields between unstructured volume and surface meshes, which can be arbitrarily distributed among different processors. No a priori information on the relation between the different meshes is required. Our inherently parallel approach is general in the sense that it can deal with both classical interpolation and variational transfer operators, e.g., the $L^2$-projection and the pseudo-$L^2$-projection. It includes a parallel search strategy, output dependent load-balancing, and the computation of element intersections, as well as the parallel assembling of the algebraic representation of the respective transfer operator. We describe our algorithmic framework and its...