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The black hole shadow, first observed by the Event Horizon Telescope in 2017, is the newest method for studying black holes and understanding gravity. Much work has gone into understanding the shadow of a Kerr black hole, including all of the complex astrophysics of the accretion disk, and there are numerous studies of the ideal shadow in non-Kerr black holes and exotic compact objects. This paper presents one of the first studies of the black hole shadow of non-Kerr black holes when the illumination source is an accretion disk. In particular, the ability of current and future very long baseline interformeters to estimate the physical parameters of the black hole spacetime and accretion disk is investigated using two different parametrized black hole metrics that encode a number of possible deviations from Kerr. Both the full shadow image and the individual subrings of the shadow are analyzed as the higher order subrings are weakly dependent on the disk physics and may be a more viable observable for studying the spacetime. The results suggest that with current telescope capabilities and any future earth-based telescopes it will be quite difficult to place strong constraints on departures from the Kerr spacetime, primarily due to the low resolution and strong degeneracies between the spacetime parameters. More optimistically, space-based interferometers may be capable of testing the Kerr nature of black holes and general relativity to comparable or better precision than is currently possible with other observations.