We measure the mobility of nanoparticles at low concentrations in non-Newtonian semidilute aqueous solutions of high-molecular-weight polyelectrolyte polymers. Using optical microscopy and particle tracking algorithms, we image and track hydrophilic polystyrene nanoparticles of diameter 400 nm moving in aqueous solutions of partially hydrolyzed polyacrylamide of molecular weight 8 000 000 Da and concentration of 0.042-4.2 g/L. The effective diffusivity of the nanoparticles in the semidilute polymer solutions, extracted from the long-time limit of the mean-squared displacement using the Stokes-Einstein relation, is greater than that calculated from the zero-shear-rate viscosity measured using bulk rheology. For concentrations c > 0.42 g/L, the mean-square displacements (MSD) of particles measured as a function of lag time revealed that the particle dynamics are subdiffusive at short time scales and are Fickian on long time scales. The time scale for the crossover from subdiffusive to Fickian dynamics increases with increasing polymer concentration; moreover, it is longer than the relaxation time scale for polymer blobs and shorter than that for the chain. Our results suggest that the nanoparticle dynamics are coupled to those of the polymers on a length scale intermediate between the blob size and the end-to-end distance of the polymer.