This is why it seems that there are not as many of them at that time of year. They roost in a small park at Front and Jackson. This park is often mistakenly called Jackson Square or MacArthur Park. Its real name is Walton Square. Not at all.
Many parrots are able to tolerate cold temperatures.
They have a down layer which will grow thicker if needed. The real issue is whether they can get enough food to eat or not. They use cavities in trees which they will expand if need be. They will not start a hole from scratch, however. Their preferred tree for nesting is the Canary Island Date Palm.