This paper introduces an adjoint Krylov method for parameter inference in large, sparse linear evolution equations whose system matrices depend affinely on unknown coefficients. We estimate these coefficients from discrete observations by constructing one forward and one adjoint Krylov approximation on each observation interval. Quadrature of the resulting bilinear form gives the full coefficient gradient through sparse matrix-vector products, without evaluating a separate Frechet derivative in every operator direction. We separate the gradient error into forward, adjoint, and quadrature terms and derive residual-based bounds for Markov and Lindblad generators. We also identify cases in which heuristic stopping estimates become small before the gradient has converged. With workspace reuse, we compute library gradients for a 12-qubit Lindblad generator with dimension 1.7 x 10^7 on a single machine with 16 GB of memory. Experiments on chemical master, Kuramoto-Sivashinsky, and Lindblad equations demonstrate accurate gradients and parameter recovery.

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