A parametric nonlinear lattice model is developed for investigating the free undamped propagation of elastic waves in two-dimensional textile metamaterials characterized by a spatially periodic prestressed configuration. The weakly nonlinear dynamics of the system are governed by differential equations incorporating quadratic and cubic terms deriving from the elastic contact between plain woven wires. By combining the multiple scale methods for harmonic motions with spatial periodicity conditions, the linear dispersion properties of the metamaterial are described, and their nonlinear perturbations are asymptotically determined as analytical function of the wave oscillation amplitude. Interesting dynamic phenomena like waveform polarization and wavefrequency softening are disclosed in the linear and nonlinear fields.