Cod, salmon, pufferfish, and tropical fish all have (blank) ambient sounds. Parrots will still attempt to mimic the fish to no avail, due to no there being no equivalent for when the parrot mimics it, unlike other mobs (barring the 1.14 mobs, see MC-138078), which have their own "parrot mimicking" sound effect.

The first few screenshots show that all 4 fish mobs do indeed have ambient sounds (presumably leftover from when the April Fools' "new resource pack" was released, and fish "sang"), and the next one shows parrots do indeed try to mimic the fish mobs. The rest of the screenshots show the list of "entity.parrot.imitate" sound list from using /playsound, showing that there is indeed no equivalent.

According to a comment by tryashtar on MC-127166, parrots shouldn't be imitating "squid, guardians, and fish" in the first place!

EDIT: New screenshots confirm that this works with ALL fish, and not just cod. Pufferfish might be WAI, due to being a neutral mob.