In this paper, the response behavior of two parallel single-degree-of-freedom structures connected with Maxwell damper is studied under base acceleration modeled as a non-stationary random process as well as stationary white-noise random excitation. The governing equations of motion of the connected structures are formulated and root mean square responses (relative displacement and absolute acceleration) are obtained. The responses are obtained considering 1000 ground motion realization using Monte Carlo simulation. The influence of parameters such as relaxation time of damper, frequency, and mass ratio of structures on the performance of damper is investigated. For undamped coupled structures, the closed-form expressions for optimum damper damping for the minimum value of mean square relative displacement and absolute acceleration of either of coupled structures are derived.