No, you shouldn’t compost fish or any leftover parts, like heads or bones.
The smell of rotting fish or other fish products is likely to attract rats and other animals to your compost heap so best to avoid adding it if possible.
(Some people with completely sealed bins or very active wormeries are happy to add it, particularly just little scraps.)
Don’t forget that fish bones and heads can be used to make fish stock as well as other meals.
Is it just becouse of the smell and the Chance of attracting rats, becouse i have read that in alot of coastal Areas in europe bycatch was used to fertilize the fields