It's not likely that your bird will get fleas, although not impossible. The best way to treat a flea infestation is to treat the mammal pets (which you have) and then treat your house. But you generally cannot treat your house with the bird in it, so you may need to do one room at a time or find a place to board your bird.
You can get some other good information on this subject if you google "can birds get fleas."
I've seen all kinds of parasites on birds (wild birds that tend to be buggy anyway): maggots, feather mites, lice and feather flies (the creepiest of all). But I've never seen a flea naturally occurring on a bird. I suppose if the house was heavily infested with fleas, they might try a bird. But then again, you would be pretty uncomfortable with that many fleas as well.
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