The credits on that page are unfortunately incomplete. Here's what it has:
Composition:
Daisuke Minamizawa: 1.01, 1.03~10
Masaru Suzuki (
this one): 1.02, 2.18, 2.41, 2.42, 2.46, 2.49
Hidekazu Hoshino (星野英和): 1.11~14, 2.01~05, 2.08~10
Kazuhiko Maeda (前田和彦): 1.15~21, 2.06, 2.07, 2.14, 2.15, 2.17
Studio Cliche: 1.22~26, 1.30~36
Gaku Hirose (広瀬学): 1.27~29, 2.11~13
CUBE: 2.19~22, 2.24, 2.25, 2.43~45, 2.47, 2.48
Noboru Iwata: 2.23, 2.30~34, 2.40
Pietro Mascagni: 2.26
John Philip Sousa: 2.27
Junzo Yagami: 2.29
Unknown composers: 1.37~40, 2.16, 2.28, 2.35~39
Arrangement:
Junzo Yagami: 1.01, 2.29
Masaru Suzuki: 1.02~07, 1.09, 2.18, 2.41, 2.42, 2.46, 2.49
Hiroyuki Kouzu: 1.08, 2.19~21, 2.24, 2.25, 2.43~45, 2.47, 2.48
Daisuke Minamizawa: 1.10
Studio Cliche: 1.11~36, 2.01~13
CUBE: 2.22
Noboru Iwata: 2.23, 2.26~28, 2.31~34, 2.40
Unknown arrangers: 1.37~40, 2.16, 2.35~39
Track 2.28 is an arrangement of a folk song, but the other composers and then arrangers of those tracks, if they exist, are up in the air. There's a
subpage that apparently lists all staff, and it includes some people who aren't accounted for:
Composition: 谷崎洋昭, 政谷剛史
Vocals: 飯田恭子, 千葉陽子, ちゃみ, 民てる代, Junzo Yagami, 黒田和歌子, 進藤香