I don't see how Caiques could be considered related to Poicepahlus. Are you just going by their similar size and colored head?
You see, Poicephalus parrots are part of the Psittacini tribe. Meanwhile Caiques are from the Arini tribe which comprises new world parrots like conures and macaws.
Parrots existed prior to the break up of Gondwanaland in the late Cretaceous period so it is highly probable that common ancestors became geographically separated during the continental divide (1). Furthermore a greater genetic divide formed between Psittacidae and Cacatuidae not only as a result of the break off of Australia from the supercontinent but also from the greater geographical separation from the South America-Africa parrots because those two continents were land locked (2).
Psittaciforme ancestry thus appears to go back about 100 million years. What is kind of freaky is that it is likely parrots existed well before T.Rex ever came about about 67 million years ago (3). Although it is unlikely that parrots lived in the same environments as Tyrannosaurs Rex at the time, it is quite likely that parrots coexisted with local land based dinosaurs in Gandwana (what is now Africa, South America, and Australia). Furthermore parrots are one of the earlier branched off lineages of birds that are still present to this day.
By 80 million years ago, you can observe a significant break up of Gondwanaland and interchange of species was becoming difficult if not impossible by this point. Remember that parrots are non-migratory birds and would not travel divides much larger than this. Therefore they settled and became separated into 3 main regions of South American, African, and Australian parrots.
Neornithes are the line of modern birds and parrots originate from the line of Neoaves. I outlined the lineage you can find parrots coming from.
So to answer your question, it is actually quite impossible that Caiques have any sort of close relation with poicephalus parrots and any similarities are the result of convergent evolution.
So when you think of what these little creatures had to live with (big scary dinosaurs) it's no wonder why they are so hardy, smart, and developed!
(1) http://mbe.oxfordjournals.org/cgi/conte ... 25/10/2141
(2) http://scienceblogs.com/grrlscientist/2 ... _of_pa.php
(3) http://www.nhm.ac.uk/jdsml/nature-onlin ... annosaurus