We consider a class of stochastic linear elasticity problems whose elastic moduli depend linearly on a countable set of random variables. The stochastic equation is studied via a deterministic parametric problem on an infinite-dimensional parameter space. We first study the best N-term approximation of the generalized polynomial chaos (gpc) expansion of the solution to the displacement formula by considering a Galerkin projection onto the space obtained by truncating the gpc expansion. We provide sufficient conditions on the coefficients of the elastic moduli’s expansion so that a rate of convergence for this approximation holds. We then consider two classes of stochastic and parametric mixed elasticity problems. The first one is the Hellinger–Reissner formula for approximating directly the gpc expansion of the stress. For isotropic problems, the multiplying constant of the best N-term convergence rate for the displacement formula grows with the ratio of the Lame constants. We thus consider stochastic and parametric mixed problems for nearly incompressible isotropic materials whose best N-term approximation rate is uniform with respect to the ratio of the Lame constants.