X-ray reflection spectroscopy is a promising technique for testing general relativity in the strong-field regime as it can be used to test the Kerr black hole hypothesis. In this context, the parametrically deformed black hole metrics proposed by Konoplya, Rezzolla, and Zhidenko [Phys. Rev. D 93, 064015 (2016)] form an important class of non-Kerr black holes. We implement this class of black hole metrics in RELXILL_NK, which is a framework we have developed for testing for non-Kerr black holes using x-ray reflection spectroscopy. We perform a qualitative analysis of the effect of the leading order strong-field deformation parameters on typical observables like the innermost stable circular orbits and the reflection spectra. We also present the first x-ray constraints on some of the deformation parameters of this metric, using Suzaku data from the supermassive black hole in Ark 564, and compare them with those obtained (or expected) from other observational techniques like gravitational waves and black hole imaging.

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