This paper presents a new parallel rheological constitutive framework intended to accurately predict the nonlinear, time-dependent response of elastomers and polymers subjected to large deformations. This framework has been implemented in the commercial finite element software Abaqus. The framework consists of an arbitrary number of viscoelastic networks and an elastic equilibrium network, which are connected in parallel. In the model the multiplicative decomposition of the deformation gradient into elastic and inelastic parts is assumed in each network. In this approach there exists an intermediate, stress-free configuration from which the elastic response is computed. The elastic response is specified using any of the hyperelastic strain energy functions available in Abaqus. The viscous response is obtained assuming the existence of the creep potential, which is used to compute the flow rule. The flow rule is expressed as a function of stress invariants and internal variables, and different evolution laws for the internal variables are allowed within the framework. Finally, a specific nonlinear viscoelastic model based on the parallel rheological framework is used to predict the response of a PC-ABS thermoplastic material for a series of relaxation tests, and the resulting numerical solutions are compared with the experimental results.