Motivated by the necessity to identify stochastic parameters in a wide range of stochastic partial differential equations, an abstract inversion framework is designed. The stochastic inverse problem is studied in a stochastic optimization framework. The essential properties of the solution map are derived and used to prove the solvability of the stochastic optimization problems. Novel convergence rates for the stochastic inverse problem are presented in the abstract formulation without requiring the so-called smallness condition. Under the assumption of finite-dimensional noise, the stochastic inverse problem is parametrized and solved by using the Stochastic Galerkin discretization scheme. The developed framework is applied to estimate stochastic Lam\'e parameters in the system of linear elasticity. We present numerical results that are quite encouraging and show the feasibility and efficacy of the developed framework.