In this paper, we first propose an adaptive bit-level data embedding (ABDE) method to embed secret data into a cover image and an adaptive checkerboard based prediction (ACBP) method to predict 3/4 of the pixels in an image using its remaining 1/4 of the pixels. Based on ABDE and ACBP, we further propose a parametric reversible data hiding method in encrypted images (PRDHEI). When parameter epsilon = 1 ( epsilon is an element of [1, 255]), PRDHEI is a full reversible method that both the original image and secret data can be completely recovered. The embedding rate is much higher than state-of-the-art RDHEI methods. Moreover, when epsilon > 1, the embedding rate of PRDHEI increases significantly. The receiver can fully recover the secret data and reconstruct the original image with very high quality. When epsilon = 255, PRDHEI reaches its maximum embedding rate of 4.5 bpp while the recovered images are of an average peak signal-to-noise ratio larger than 32 dB. (C) 2018 Elsevier B.V. All rights reserved.