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 micromechanics-informed parametric DMN (MIpDMN) architecture is proposed for multiscale materials with a varying microstructure characterized by several parameters. A single-layer feedforward neural network is used to account for the dependence of DMN fitting parameters on the microstructural ones. Micromechanical constraints are prescribed both on the architecture and the outputs of this new neural network. The proposed MIpDMN is also recast in a multiple physics setting, where physical properties other than the mechanical ones can also be predicted. In the numerical simulations conducted on three parameterized microstructures, MIpDMN demonstrates satisfying generalization capabilities when morphology varies. The effective behaviors of such parametric multiscale materials can thus be predicted and encoded by MIpDMN with high accuracy and efficiency.