Deep Material Network (DMN) has recently emerged as a data-driven surrogate model for heterogeneous materials. Given a particular microstructural morphology, the effective linear and nonlinear behaviors can be successfully approximated by such physics-based neural-network like architecture. In this work, a novel parametric DMN architecture is proposed for multiscale materials with a varying microstructure characterized by several parameters. A Physics-Informed Neural Network (PINN) is used to account for the dependence of DMN fitting parameters on the microstructural ones. Micromechanical constraints are prescribed both on the network architecture and on the output of this PINN. The proposed PINN-DMN architecture is also recast in a multiphysics setting, where physical properties other than the mechanical ones can also be predicted. In the numerical simulations conducted on three parametric microstructures, PINN-DMN demonstrates satisfying interpolative and extrapolative generalization capabilities when morphology varies. The effective multiphysics behaviors of such parametric multiscale materials can thus be predicted and encoded by PINN-DMN with high accuracy and efficiency.