Reversible data hiding (RDH) plays a key role in ensuring security of communications. Recent studies have revealed that RDH techniques based on interpolation are gaining popularity because of the ease of implementation and support for high embedding capacity. From recent literature, it is noted that the one-dimensional (1D) parabolic interpolation-based data hiding technique is suitable for high embedding capacity applications. This study aims to implement a high capacity RDH technique using a new two-dimensional (2D) parabolic interpolation and a novel embedding technique that is suitable for interpolation-based techniques. The results of this study showed that the proposed 2D parabolic interpolation maximized the utilization of the cover image pixels in up-sampling. The technique produced interpolated images of a higher quality compared to the 1D parabolic interpolation technique. Moreover, the proposed data hiding technique exploited the local redundancy and obtained the high embedding capacity with an appreciable image quality better than other state-of-the-art techniques. The results of this study support the view that the proposed parabolic interpolation has the potential to address security issues in high embedding capacity applications. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.