Grafting polymer chains on the surface of nanoparticles (NPs) is a strategy used to control the interaction between the NPs and their environment. The fate of the resulting particles in a given environment is strongly influenced by the solvent-polymer interaction. The solvent quality affects the behavior, conformation, and dynamics of the grafted polymer chains. However, when this polymer grafting strategy is used to functionalized polymer particles, the influence of solvent quality becomes even more complex; when the grafted polymer chains and the polymer nanoparticles are tethered together, the effect of the solvent quality on the behavior and dynamics of the system depends on the solvent interaction with both polymer components. To explore the relationship between the solvent quality and the dynamics of polymer-functionalized soft polymer NPs, we designed a system based on cross-linked polystyrene (PS) NPs grafted with a canopy of poly(methyl acrylate) (PMA). PS and PMA, two immiscible polymers, can be selectively solvated by using binary mixtures of solvents. NMR spectroscopy was used to address the effect of those selective solvents on the local mobility of the PS-PMA core-canopy NPs and revealed an interplay between the local mobility of the core and the local mobility of the canopy. A selective reduction of the solvent quality for the PMA canopy resulted in the expected reduction of the local mobility of the PMA chains, but also in the slower dynamics of the PS core. Similarly, a selective reduction of the solvent quality for the PS core resulted in a slower dynamics for both the PS core and the PMA canopy.