Steganography is a technique applied to ensure secure communication. It is challenged by visual observation or statistical analysis to ascertain whether a message is hidden within a cover medium. Several steganographic methods have been proposed for palette-based images. These methods maintain image quality but cannot resist some statistical and visual attacks. To overcome this problem, two parity assignments with performances similar to those of existing parity assignments are proposed. An innovative embedding process that randomly selects one parity assignment while embedding secret bits in each pixel is provided. Finally, a steganographic method using the proposed embedding process and a specified adaptive scheme is presented. Experimental results revealed that the proposed method is undetectable under some statistical and visual attacks even as it maintains image quality.