If a bird does lots of nest buildung in the wild and you have a big enough cage. Is it a good idea to breaks some twigs off a tree for them to build while you're out?
Parrots generally use hollowed cavities as nests rather then building structures as a crow would do for example. There might be the odd exception (I think quaker parrots might build them? ) but generally if you give them a load of sticks they'll just chew them.
It's also worth bearing in mind that birds build nests for the purpose of laying eggs and bringing up babies rather than as a house for themselves - so for a pet bird where egg laying is a serious potential health risk this isn't a terribly good thing to encourage anyway.