Also the "action" genre is a very broad superordinate genre that includes any subordinate genre that isn't turn-based or similar I think (platformers, shooters and even action RPGs). I don't like the "arcade" genre so much either since it's way too broad for a genre. But if it would count as a genre it could be considered a superordinate genre that includes any genre that would fit in an arcade hall I guess. Which would be about anything but RPGs and adventure games, but nowadays there are arcade machines that save your progress on a card that you bring with you, so even those kind of games could be crammed in, technically speaking.
I agree, I find both Action and Arcade to be pretty generic descriptors myself, but there is a degree of nostalgia to using either of them that works for me in some cases. I hate struggling over fitting things into genres (organizing my music is one place where I have tons of trouble as I'm sure do lots of people with eclectic or just widely varied tastes), but there are definitely games that don't fit well into the established conventions we have today. "Action" makes me think back to the labels from the original NES releases, so there is a degree of nostalgia of using it like that. I feel like it's the "catch all" genre for when you can't really give something a solid classification otherwise. I grew up with arcade games like many members here did, so sometimes that can also feel appropriate. Parasol Stars definitely has that vibe, no surprise with it being developed by Taito, and it's even blood with a pretty famous arcade series - Bubble Bobble, of course. Doh from Arkanoid also makes an appearance, to boot. For younger members the term might not be as evocative, and I think it has become a little generalized over the years (Xbox Live Arcade, for example). I am sure there is a better description for Parasol Stars, but I don't feel platformer really covers it either. Anyway, that's just how I tend to think of things.
It's funny really how much you can differentiate from what you are looking at vs. what is happening.. for instance, Donkey Kong 3 feels like a shooter/shoot em' up to me, almost like Galaga, really. It's just instead of a spaceship you have a guy working in a greenhouse with a bug sprayer. Guardian Heroes lets you traverse between 3 planes in the foreground, middle ground, and background. (Or Yuu Yuu Hakusho MD, if you prefer). The genre isn't the same at all, but it's kinda like what you are doing in Parasol Stars - moving up and down between "platforms" to deal with dispatching enemies. Of course I will still classify Guardian Heroes as a Beat 'em up ala Final Fight and it's ilk, but it's funny to think about how similar some of the gameplay elements can be if you could just change the artwork.