In this paper, an N-parallel FENE-P constitutive model based on multiple relaxation times is proposed, it can be viewed as a simplified version of the multi-mode FENE-P model under the assumption of identical deformation rate. The proposed model holds the merit of multiple relaxation times to preserve good computational accuracy but could reduce the computational cost, especially in the application of high-fidelity numerical simulation of viscoelastic turbulent drag-reducing flow. Firstly the establishment of N-parallel FENE-P model and the numerical approach to calculate the apparent viscosity are introduced. Then the proposed model is compared with the experimental data and the conventional FENE-P model in estimating rheological properties of two common-used viscoelastic fluids to validate its performance. This work is an extended version of our ICCS conference paper . To further judge the performance of the proposed FENE-P model in complex turbulent flows, the extended application of the proposed model in large-eddy simulation of viscoelastic turbulent drag-reducing channel flow is carried out.