Bringing my parrots to Ecuador

My husband and I want to relocate to Ecuador in the near future, and we have 8 parrots.  I realize that we are allowed to bring only four (two each).  How difficult was it?  We would be bringing a Solomn Islands eclectus, a Goffin's cockatoo, either a Meyer's parrot or a cockatiel and a quaker (or sometimes called a monk) parakeet.  It's the quaker that I'm most concerned about since they originate in S. America and none of the others do.  However, I can't find out if they are native to Ecuador.  All the information I've found says that they are native to S. America, ranging from central Bolivia to southern Brazil, Uruguay, and southern and central Argentina, saying nothing about Ecuador.  She is my baby and I have to take her!  I've had her for 8 yrs. since she was a chick.  I dread the thought of traveling with 4 of them as we could only keep 2 in the cabin since their carriers have to fit under our seats.  They are never locked up at home either, so they are not going to be happy, but the thought of leaving without them is unthinkable.  Do you know if the ban from the bird flu has been lifted?  Any and all info you could give me would be GREATLY appreciated!  Thank you in advance.