We perform molecular dynamics simulations and develop a theoretical approach based on the two-dimensional cylindrical cell model to investigate the salt-dependent interactions between two sparsely-grafted, rigid polyelectrolyte brushes. Extending our previous study, (A. Wynveen and C. N. Likos, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 010801), we find that the repulsive force between the brushes arises in equal parts from the compression of osmotically-active counterions trapped within the brushes, and from the necessary distortion of the rigid polyelectrolytes as the brushes approach each other. This latter, bending-force contribution also depends on the ionic environment within the brush as ionic screening reduces the effective persistence length of the polyelectrolyte chains. Our investigations yield results that are consistent with those of experimental studies of the salt-dependent forces between DNA-grafted colloids.