We propose a new approach for inverse parameter estimation using Variational Graph Neural Networks (VGNNs). Our model uses a graph-based variational decoder to obtain the spatial probability distributions of the parameters, relying only on the displacements observed in the experiments. The VGNN framework naturally captures the complex, non-linear relationships between displacements, material properties, and external forces. This allows for efficient parameter estimation even when training data is limited. We applied this method to two different problems. First, we estimated the non-linear distribution of the elastic modulus in a plane elasticity problem. Second, we estimated the position and magnitude of a load applied to a 3D hyperelastic beam. We show that the inferred statistics of the elastic modulus—such as the mean, standard deviation, and spatial correlations—match the real reference distribution very well in the 2D case. Similarly, the estimates for the load position and magnitude in the 3D case are very close to the true values. These results show the precision and robustness of our VGNN approach, even when dealing with non-linear hyperelastic behaviors and complex material properties. This demonstrates the method's potential for solving difficult inverse problems in solid mechanics, suggesting practical applications in structural engineering and simulation with limited data.