The PVA gels obtained by freezing/thawing cycles of PVA solutions, also called cryogels, exhibit non-linear elastic behavior and can mimic, within certain limits, the behavior of biological soft tissues such as arterial tissue. Several authors have investigated the effects of cryogenic processing parameters on the Young’s modulus. However, an elastic modulus does not describe the non-linearity of the cryogel’s stress–strain response. This study examines the non-linear elastic response of PVA cryogel under uniaxial tension and investigates how processing parameters such as the concentration, the number of thermal cycles, and the thawing rate affect this response. The relationship between the coefficients of the material model and the processing parameters was interpolated to find the set of parameters that would best approximate the elastic response of healthy porcine coronary arteries under uniaxial tension.