The Flintstones (1988)

The Flintstones is an action game, which had been developed by Tegue Interactive for Grandslam Entertainment and released on the Sinclair ZX Spectrum, Amstrad CPC, Commodore 64, MSX and Commodore Amiga in 1988, and later for Sega, when released on their Master System in 1991; it is based upon the Hanna-Barbera television series of the same name, which aired from 1960 to 1966, and the fact that Peebles is present in the game also suggests that it is set in early 1963 (as Bamm-Bamm, who was left upon the Rubbles' doorstep later in that same year in the show, is not present). It's also the second title in the The Flintstones series - the first one being Yabba-Dabba-Doo (1985).

Story
This is from the back of the cassette's, or disk's in the case of the Commodore Amiga version, inlay card, the front of which is shown in the image to the right (it also features Betty, who is not present):

However, this is off the back of the box of the SMS's cartridge, which was released three years later and looks identical to the other versions' inlay cards apart from having the Sega logo at the top of it: