Seismic properties play a fundamental role in the geological and petrophysical modeling of reservoirs due to their dependence on petrophysical properties. Most existing stochastic seismic inversion methods are based on Gaussian probability distribution functions and assume linear dependence. Examples include sequential Gaussian co-simulation (SGCS) and direct sequential simulation (DSS). In contrast, spatial stochastic co-simulation methods based on Bernstein copulas (BCCS) have recently been developed. These methods do not require a specific distribution type or linear dependence, thereby overcoming the limitations of traditional approaches. In this context, we propose a novel approach for the joint seismic inversion of elastic and petrophysical properties using parametric copulas within a Bayesian inference framework. A joint probability distribution is constructed using well-scale petrophysical and elastic property data, fitted to parametric copula functions and treated as prior information. The model parameters are then updated a posteriori using petrophysical properties scaled by a moving window averaging method and seismic properties upscaled using the Backus averaging method. The resulting posterior model is used within the inversion process to generate elastic property realizations at the seismic scale. The inverse problem is solved using a simulated annealing algorithm that minimizes a global objective function combining the root-mean-square (RMS) error between synthetic and observed seismic traces, and the semivariogram error between the simulated and target variogram models. For each elastic realization, a reflectivity series is computed and convolved with a seismic wavelet to generate a synthetic seismic trace. The best-fitting elastic realization is then used to simulate the corresponding petrophysical property using the same joint probability distribution. The proposed method was applied to a deepwater reservoir case study to estimate total porosity and acoustic impedance at the seismic scale. Results demonstrate that the use of parametric copulas reduces computational cost and execution time while enabling effective integration of nonlinear dependencies. The synthetic traces exhibit RMS errors below 8%, validating the accuracy and robustness of the copula-based inversion framework