Loulou is the only good thing to happen to Félicité in her entire life. Just look at his entrance in the story:
That day, something wonderful happened: at dinner time, Madame de Larsonnière's n**** servant came to the door, holding the parrot in his cage, complete with perch, chain and padlock. (3.94)
Félicité is thrilled to have him, especially in her loneliness after so much death.
Loulou represents a few things for Félicité. For one thing, he's a connection to her dead nephew:
He had long fascinated Félicité, because he came from America. The word reminded her of Victor, and she had gone so far as to ask the n**** about him. (3.95)
She believes that the black servant will know something about Victor because he dies in Cuba, which she pictures as being full of only black people. Maybe she imagines that Loulou knows Victor, as well.
The bird also just becomes a friend to Félicité, something she really doesn't have in her life. People treat Loulou as a curiosity, something to be mildly tortured. Fabu, the butcher's boy, teaches him swear words and threatens to twist his neck. Meanwhile, "Monsieur Paul was careless enough to blow smoke in his nostrils" (4.7). Félicité, on the other hand treats him like a child. She protects him from the swear words by giving him "the run of the house" (4.6).
Loulou also represents all of the loss in Félicité's life. One day, "She had put him down on the grass to cool him and gone away for a minute. When she returned, the parrot was gone!" (4.8) Poor Félicité searches for him frantically, probably thinking of Victor, Virginie, even Théodore…but she can't find him.
She finally gives up and "returned, exhausted and with a heavy heart, her slippers torn to shreds. She sat down on the bench next to Madame, and was telling her all she had done to find the bird, when something light came to rest on her shoulder—Loulou!" (4.8) This is the only time that something Félicité loses comes back to her, which just strengthens her bond to the pretty bird.
Loulou finally takes on holy proportions at the end of Félicité's life. After he dies she has him stuffed, and puts him in her room with all of her other religious objects. She begins to almost worship him because of a certain resemblance he has with the Holy Spirit:
In church, gazing at the Holy Spirit, she noticed that he had something of the parrot about him. The resemblance was even more striking in a print depicting the baptism of Our Lord. With his purple wings and emerald body, the Holy Spirit was the image of Loulou.
[. . .] They were associated in her mind, the parrot being sanctified by this association with the Holy Spirit, who himself became more vivid and intelligible to her. (4.32-33)
Félicité even turns toward the bird when she prays. Loulou has become an idol, a representative of God here on earth. She has always been kept away from the divine; she has no education and can't read, and must get her religious training by taking Virginie to church. Loulou is like proof that God would care about someone as simple and poor as Félicité.
When she finally dies, Félicité has her ultimate encounter with Loulou:
Her heart slowed, becoming vaguer and gentler with each beat, like a fountain running dry, like an echo fading away, and as she breathed her last, she seemed to see, hovering above her head as the heavens opened, a giant parrot. (5.13)
It's almost comical, but the absurdity of a parrot as the Holy Spirit is the whole point; Félicité has to create her own relationship with God because of her social status.