Classical adjoint-sensitivity analysis provides an elegant framework to efficiently compute the first-order sensitivities of a few circuit performances with respect to many circuit parameters. However, the computed sensitivity qualities are incremental in nature, hence only reflecting the performance changes under small perturbations of circuit parameters. In this paper, we rigorously extend the classical adjoint-sensitivity analysis and provide an efficient formulation for computing time-domain second-order adjoint sensitivities for linear circuits. This new formulation not only computes the exact second-order sensitivities but also has a linear complexity in the number of parameters. Specifically, it only takes P+2 transient runs to compute all O(P2) first- and second-order performance sensitivities, where P is the number of parameters. Circuit examples are included to demonstrate the proposed formulation under the context of parametric circuit analysis and optimization.