Parasol Stars was Bubble Bobble 3, for the Turbo Grafx-16/PC Engine platform.
In my opinion, it's one of the best of the series.
Instead of using bubbles to trap enemies, you trap them on your umbrella and throw them at others.
If you have a TG-16, the game is still available direct from Working Designs.
The Parasol Stars Box Art
More box art from another system
The Parasol Stars FAQ