Bubble Bobble is not only one of the most popular arcade ever created, but also a personal favourite of GamesNostalgia. This 2D cartoon action platformer designed by Fukio Mitsuji (aka MTJ) was released by Taito initially in 1986.
Rainbow Islands is the sequel to Bubble Bobble and was developed and published by Taito in 1987 for arcade only. It would remain arcade only until 1990 when it was ported to the Amiga, Commodore 64 and others.
Parasol Stars, also known as Parasol Stars: The Story of Rainbow Islands II - or if you prefer, The Story of Bubble Bobble III (not sure why, since there was no "Story of Bubble Bobble I").