The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field partial differential equation model with the Young modulus represented as an affine function of a countable set of parameters. We introduce a weak formulation, establish its stability with respect to a weighted norm and discuss its approximation using stochastic Galerkin mixed finite element methods. We motivate an a posteriori error estimation scheme and establish upper and lower bounds for the approximation error. The constants in the bounds are independent of the Poisson ratio as well as the spatial and parametric discretisation parameters. We also discuss proxies for the error reduction associated with enrichments of the approximation spaces and we develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. The error reduction proxies are shown to be reliable and efficient in the incompressible limit case. Numerical results are presented to supplement the theory. All experiments were performed using open source (IFISS) software that is available online.