The birds not only represent different ideas but they also represent different characters.
The parrot in Chapter One can be compared with Edna. She too has fine feathers and is kept effectively in a cage. Her role in society is to look nice but her freedom is also as limited as the parrot's.
On the Isle, birds are free and are not caged as they are in city. However, the next specific mention of birds is in connection with Edna's new house nicknamed the pigeon house. It is no coincidence because she feels as cooped up as a kept pigeon would.
Madame Reisz sees Edna as a bird too afraid to fly and Edna is also aware that her friend sees her so.
In the end, we have the image of a bird with a broken wing. Here we have a bird (Edna) who wants to fly but now cannot. Even though she has finally been able to get up enough courage to try and fly, her attempt is thwarted by a broken wing. In essence, this symbolizes her role in society, that of a victim who is powerless to escape regardless of her self discovery and awakening into what her life could be like.
In these examples, you can see how Kate Chopin used birds, specific kinds as well as just the image of a bird, to tell Edna's story.