Most a-vets will not chip any bird that is cockatiel sized or smaller, which usually includes the pyrrhura conures, however they will microchip birds that are quaker or sun conure sized and larger.
Pros?
The chips are small and "easily" fit underneath the skin
They can hold enough data on them to help return your lost pet
Cons?
There's been at least one report of a macaw that started mutilating himself after being microchipped
Microchipping in parrots is still relatively new, so not everyone will think to scan a lost parrot for a chip.
Not all scanners will read all chips
The majority of people who find dogs and cats usually have the mentallity to look for their owners. Although many people who find birds will do the same, I've noticed a larger portion will decide to keep the bird rather than looking for the owner... then again, I frequent bird forums far more than I do cat or dog forums, so that information could be wrong.
I'm sure there are more pros to microchipping, and that the pros probably outweight the cons. Here's a couple of articles on the subject.
http://www.birdtheft.org/birdid.htmhttp://www.birdsnways.com/wisdom/ww7eiii.htm