This paper identifies a new mechanism for amplification of stochastic disturbances in channel flows of strongly elastic polymer solutions. For streamwise constant flows with high elasticity numbers μ and non-vanishing Reynolds numbers Re, the O(μRe3) scaling of the variance amplification is established using singular perturbations techniques. This demonstrates that large variances can be maintained in stochastically driven flows occurring in weak inertial/strong elastic regimes. Mathematically, the amplification arises due to nonnormality of the governing equations and, physically, it is caused by the stretching of the polymer stresses by the background shear. The reported developments provide a possible route for a bypass transition to 'elastic turbulence' and suggest a novel method for efficient mixing in micro-fabricated straight channels.