Existing online continuous-time parameter estimation laws provide exact (asymptotic/exponential or finite/fixed time) identification of dynamical linear/nonlinear systems parameters only if the external perturbations are vanishing to zero or independent with the regressor of the system. However, in real systems the disturbances are almost always nonvanishing and dependent with the regressor. In the presence of perturbations with such properties the abovementioned identification approaches ensure only boundedness of a parameter estimation error. The main goal of this study is to close this gap and develop a novel online continuous-time parameter estimator that guarantees the exact asymptotic identification of unknown parameters of linear systems in the presence of unknown but bounded perturbations and has relaxed convergence conditions. To achieve the aforementioned goal, it is proposed to augment the deeply investigated dynamic regressor extension and mixing (DREM) procedure with the novel instrumental variables (IV)-based extension scheme with averaging. Such an approach allows one to obtain a set of scalar regression equations with asymptotically vanishing perturbation if the initial disturbance that affects the plant is bounded and independent not with the system regressor, but with the IV. It is rigorously proved that a gradient estimation law designed on the basis of such scalar regressions ensures online unbiased asymptotic identification of the parameters of the perturbed linear systems if some weak independence and excitation assumptions are met. Theoretical results are illustrated and supported with adequate numerical simulations.