Transavia Airlines is owned by KLM Airlines and is their means of competing in the budget, low cost transport with in Europe. It is a no frills, cheap and dirty, barebones airlines.
KLM is one of my favorite airlines and I have many hours flying them internationally.
As a general statement, transporting a Parrot within Europe demands a ton of paperwork and lots of time. Regardless of the airlines you use, this will not be easy to make happen!
I am not going to spend a great deal of time working you though this process if you do not have at least six months prior to your need to move as that time line is a reality, especially if you are attempting to make this happen from someplace outside of Europe.