This technical note addresses the gradient-based parameter estimation problem for pairwise linear Gaussian systems. The new adaptive filtering scheme is based on gradient-based optimization methods for estimating the uncertain system parameters and a robust square-root variant of the pairwise Kalman filter used for estimating the unknown states of the pairwise linear Gaussian systems. Hence, in the adaptive filtering techniques, the dynamic state and system parameters are estimated simultaneously. The method of the filter sensitivities computation required in gradient evaluation is derived in terms of square-root factors of covariance matrices.