I feel like categorizing a game as a platformer, at least in my head, denotes some trickiness or danger from jumping between platforms. Super Mario Bros. is more of a platformer than Mario Bros. because the former has difficult jumps that can be missed, pits, spikes, and other dangers in the floor that will kill you when you miss your mark. In Mario Bros. and in Parasol Stars too, there are platforms but they exist only in order to facilitate you clearing the screen of enemies - you can't die by using them and there is no inherent difficulty in traversing them. However, in Donkey Kong, you can die by falling off the platforms - and barrels and springs are always tripping you up to make you do just that. Even though it's only a single screen game, it meets my above criteria for platformer more than Parasol Stars does. The platforming element is pretty prevalent in DK. One of my favorite C64 games, Impossible Mission, is most definitely a platformer (it's also single screen / non-scrolling). Every room is a challenge where jumping could easily throw you into a robot (who kill you on contact) or through a hole in the floor.
I guess all I'm getting at is the platforms themselves should play a more important role to the gameplay before calling any game with platforms in it a platformer. That's just how I think about it though, and what makes sense to me. I tried to think of a scrolling game with innocuous platforms (won't cause you to die and/or tricky in a non lethal way) but I can't come up with one at the moment. I'm also sure there are some exceptions you guys can think of too, but for me Parasol Stars is an amazing Action/Arcade game.
Edit: 1st vote for JJ & Jeff! C'mon peeps, now that is a hardcore platformer! The more I think about it, the more I wanna get a complete box and manual for my decades old loosie. Should pick up a Sparky label and a repro box from pcenginefx when he produces them again!