It is recommended:
a) because you are rehoming an animal that probably went trough a lot and, hopefully, giving them the love they deserve.
b) taking care of a parrot properly is very difficult, hence a lot of people give them to sanctuaries that are full to the brim
c) decreasing the demand of these species. Both their decrease in their native range and their presence in non native ranges is, even now, caused by the demand of them as pets. Poaching in the Central and Southern America is incredibly high (in certain areas of Honduras is 100% for example) and they are poached to be sold to middlemen that will import them to the rest of the world.
Now, since CITES, it’s difficult to find poached animals in NA and EU, but definitely not impossible. The main market is Asia, where they can be sold locally or then transferred to the west.
At the same time, aviaries or “parrot mills” (some people call them like that) are the only reason why we have invasive parrots. If a couple of non native species get released by a distracted owners, the couple just ends up dying because they are not used to the environment. If a stock of hundreds or even thousands of animals of the same species escape an aviary they will first become a colony, then an invasive species. This phenomenon is called propagule pressure.
Invasive species destabilise the natural balance, and invasive species of parrots are stealing nesting sites of owls and bats already.
This is why we should at all cost, decrease the demand of them.