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The estimation of racial disparities in various fields is often hampered by the lack of individual-level racial information. In many cases, the law prohibits the collection of such information to prevent direct racial discrimination. As a result, analysts have frequently adopted Bayesian Improved Surname Geocoding (BISG) and its variants, which combine individual names and addresses with Census data to predict race. Unfortunately, the residuals of BISG are often correlated with the outcomes of interest, generally attenuating estimates of racial disparities. To correct this bias, we propose an alternative identification strategy under the assumption that surname is conditionally independent of the outcome given (unobserved) race, residence location, and other observed characteristics. We introduce a new class of models, Bayesian Instrumental Regression for Disparity Estimation (BIRDiE), that take BISG probabilities as inputs and produce racial disparity estimates by using surnames as an instrumental variable for race. Our estimation method is scalable, making it possible to analyze large-scale administrative data. We also show how to address potential violations of the key identification assumptions. A validation study based on the North Carolina voter file shows that BISG+BIRDiE reduces error by up to 84% when estimating racial differences in party registration. Finally, we apply the proposed methodology to estimate racial differences in who benefits from the home mortgage interest deduction using individual-level tax data from the U.S. Internal Revenue Service. Open-source software is available which implements the proposed methodology.