The finite-volume direct averaging micromechanics (FVDAM) theory for periodic heterogeneous materials is extended by incorporating parametric mapping into the theory’s analytical framework. The parametric mapping enables modeling of heterogeneous microstructures using quadrilateral subvolume discretization, in contrast with the standard version based on rectangular subdomains. Thus arbitrarily shaped inclusions or porosities can be efficiently rendered without the artificially induced stress concentrations at fiber/matrix interfaces caused by staircase approximations of curved boundaries. Relatively coarse unit cell discretizations yield effective moduli with comparable accuracy of the finite-element method. The local stress fields require greater, but not exceedingly fine, unit cell refinement to generate results comparable with exact elasticity solutions. The FVDAM theory’s parametric formulation produces a paradigm shift in the continuing evolution of this approach, enabling high-resolution simulation of local fields with much greater efficiency and confidence than the standard theory.