Computational mechanics increasingly demands modeling frameworks that are not only accurate and scalable but also capable of providing reliable uncertainty quantification. This has led to growing interest in operator-learning techniques for probabilistic solutions to parametric differential equations. In this work, we propose an uncertainty-aware operator learning framework based on Gaussian Processes (GPs) that effectively combines the expressive power of neural operators with the probabilistic nature of GPs. The central idea of the proposed approach is to construct a neural operator–embedded kernel, where a standard GP kernel is defined in a latent space learned by a deterministic neural operator. This formulation enables the GP to operate on function spaces in a resolution-invariant manner. To jointly optimize the neural operator parameters and the GP hyperparameters, we employ a stochastic dual descent (SDD) algorithm. This training strategy addresses two major limitations of classical GP models: dependence on input discretization and prohibitive cubic computational complexity. As a result, the proposed framework achieves scalable learning and prediction for high-dimensional and nonlinear parametric systems commonly encountered in computational mechanics. The method is validated across a range of nonlinear parametric partial differential equations, demonstrating superior predictive accuracy, improved computational efficiency, and well-calibrated uncertainty estimates compared to conventional Gaussian Processes and neural operators. Overall, the proposed neural operator–induced Gaussian Process framework provides a robust and scalable solution for uncertainty-aware operator learning, making it particularly suitable for complex computational mechanics applications involving limited or noisy data.