What do Birds of Paradise do?
For what I've noticed, they are able to collect coins for you when flying through them. It's something like a weaker magnet powerup. The rest of their signigficance, if any, remains unknown to me though.
What is worth checking, but might as well be not applicable:
• they might give you points to your current combo when they start flying
• and/or they might give you points to your combo when they depart (think of it as successfully riding with them for the whole distance they fly with you)