Weird as this may sound, there are a few games out there that try to put shoot-em-up into their perfectly innocent side-scrolling action platformer or action RPG. Here's the only two I can think of right now, but feel free to suggest others.
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You could skip them. I thought they were pretty fun, but I would have preferred either more depth, or just a more strategy-oriented version of this to complement the rest of the game's design. Pretty wacky and neat, though, and a nice diversion. Just wish it had a full mode that you could play straight through, like an arcade-style interface.
You know how it is: you're fighting demons, but they're like, flying away. So you, like, sprout wings all of a sudden, then shit gets real.
Flight of the Pullet! Probably one of the better games-in-games I remember playing. You have to wonder just how many eggs the titular protagonist could carry in her special place, though.
Suggested by Brackynews for its Space Invaders style sequence.
The SHMUP here seems a bit less hidden, but it's still part of a bunch of different styles of gameplay.
Includes sub and plane side-scrolling battles, though far too infrequent for them to feel integral (yet, as with others on this list, one of the SHMUPS winds up being the way you combat the end-game boss).
Not sure if this counts, but if I expand to strangely shoehorned action sequences then this and many others could easily be part of the pantheon of non-standard gaming sequence deities.
Not exactly hidden, but there are several missions that are perfectly legitimate ways to get experience and sometimes loot, which have you blowing stuff up, Afterburner style.
Surprising amount of arcade-style sections in this game. Sometimes literally!
Not so hidden, once you know about this franchise's tendency for all sorts of different game mechanics.