A formulation is developed, based on the adjoint governing equations, for efficiently computing the spatially dependent sensitivity of a quantity of interest in a flow to the parameters of a rheological model for the fluid. It is demonstrated for the viscous resistance of a sphere moving within a fluid-filled round tube, both for a Carreau model for Carbopol and a four-parameter Moore model for a generic thixotropic fluid. The field sensitivity information provided is valuable for quantifying the relative importance of different underlying physical mechanisms of the fluid and their respective roles in different regions of the flow, for directing rheological characterization to improve fidelity for a particular prediction, for inferring rheological properties from indirect measurements, and in aiding the design of fluids to meet objectives.