I'm sorry for your loss, as others have mentioned it's always so hard to say goodbye to a pet who is not only a part of your family but an integral part of your everyday life. It's not easy to go through your day when there are so many reminders of things that you enjoyed with your parrot, but if you can focus on the positive, remember the good times and fun you were able to share, it can make it easier. Parrots can be such amazing birds, they are wonderful communicators and express so many feelings that we can easily identify and relate too, think about how you were lucky to be a part of that, to share in a relationship that was so enjoyable for the two of you. Clearly it was a relationship that you were both better for being a part of, she had someone that loved her and cared enough to take such special care in her arrangements, for a parrot I can think of nothing better than having that one person they know loves them.
Like you I am not religious, I identify as atheist as well, when I think of death and loss I try not to think of it as an end but simply as a transformation from one state to another, her energy and the particles that made her who she is are still a part of this world. She may now begin another journey, becoming something else, a part of her will live on in the wind that rushes by on a warm autumn day, she may become a part of a tree bringing new life into the world or she may become the star stuff we all started as. She's still here, just not in the organized form you knew her as.