The structure and function of polymers in confined environments, e.g., biopolymers in the cytoplasm, are affected by macromolecular crowding. To explore the influence of solvent quality and dimensionality on conformations of crowded polymers, polymers are modeled as penetrable ellipsoids/ellipses, whose shapes are governed by the statistics of random walks. Within this coarse-grained model, Monte Carlo simulations of two and three-dimensional polymer-nanoparticle mixtures, including trial changes in polymer size and shape, are performed. Penetration of polymers by nanoparticles is incorporated via a free energy cost predicted by polymer field theory. Simulation results of polymer conformation are compared with predictions of free-volume/area theory for polymers in good and theta solvents. Results indicate that dimensionality and solvent quality significantly affect crowded conformation, especially in the limit of small crowders. This approach may help to motivate future experimental studies of polymers in crowded environments, with relevance for drug delivery.