Parodius is fun. Even if you already own the arcade-perfect PS1/SAT Deluxe pack version, the SNES version will offer you some bonus content.
I usually like R-Type, and thus had to have R-Type III. But eventually I found this game to be kind of a letdown. The visuals are great and pretty smooth for SNES standards, the music is so nice (especially the opening sequence and level 1), but what comes after the first level is an unfair hard unplayable mess... at least to me, since I suck at shooters where you have no "EASY MODE" option. Really, even the original R-Type and R-Type II were easier for me on normal difficuilty than part III. I had to cheat in III by using the Retrode and playing in an emulator with quicksaves, to regain some fun for the money I spent on the cartridge. My opinion? III is highly overrated.
That said, the only R-Type I've ever beaten was Super R-Type on easy and novice modes. :/
I can recommend Axelay. The Mode 7 graphics for the vertical levels feel awfully lot like being conveyor belts, but they are still somewhat impressive. The collision detection in the vertical levels need some time to get used to, but in the horizontal levels it is just fine. Good music, good weapon system.
Even though I'm a big fan of Macross, I can only think of Scrambled Valkyrie being a B-movie shooter. The graphics are OK (sometimes with cool effects), the level design is OK, the power up system is OK, but the music is horrible in some parts. Watch a longplay of some stages and you will know what I'm trying to say. Most bosses aren't taken from the movie or TV series, sadly.