Parodius

[Klatrymadon]

Aye, that's a Sexy Parodius thing that ties in with its story (which I can't remember anything about, but basically the level objectives are odd jobs that you're doing for a succession of screwball clients). Your success or failure determines whether you continue on your current route or drop down to the presumably 'bad' one. The levels are fantastic either way, so it's well worth working towards seeing them all.

[Klatrymadon]

Any time, chap. Cheers!

[DJ Incompetent]

Very good sir.
Here's some unique identifier info on how to describe each game from personal experience. I own the whole series on PSX cept the pachinko machine, the first one, and paro wars.

The first version and only "remade not straight ported" portable versions = Parodius
Longest game length and most like Gradius II = Parodius Da!
Best presentation values (graphics, music, flow, etc.) = Gokujō Parodius! (Ultimate Parodius)
Most [scoring & shit to do] depth, most selectable characters, and the game won't shutup = Jikkyō Oshaberi Parodius (Chatting Parodius)
Multiple Paths/Objectives + PG-13 Suggestive images = Sexy Parodius
Strategy game [SRPG] (something like advance wars) = Paro Wars
Literally a pachinko machine = CR Gokujō Parodius

[Drum]

If you have a PSP, import the Parodius Collection - it's super great.
I like Sexy Parodius, but the missions are the worst thing about it. DJ Incompetant's thoughts mirror my own on the series overall.
I did not know about Paro Wars, now I gotta go check that out.