Fleas are difficult, I'm sure you know...the problem with not using anything and just bathing the animals is not only the eggs, but it only takes a couple of stray fleas to make thousands...I'm not a fan of any of the spot-on stuff either, many dogs and especially cats have gone into a constant-seizure-state from all of the them, especially the cheap brands like Hartz, Seargent's, Pet Armor, etc., so never use any of them...
They do make a natural product for both fleas and ticks that I've found to work extremely well for killing ticks, and keeping fleas off of my dogs, as I've not ever seen a flea in my house, and I live in State College, PA, the freaking Deer-Tick capital of the world (no joke, it's a serious issue here)...It's main ingredient is Eugenol (Clove-Oil), and you can buy the spot-on form, the shampoo form, the spray, the collar, or the powder form, and it's leave ticks floating in the bath water...and I have a collar on both of my dogs and it's worked perfectly, and is certainly safe for the birds...I buy it at either Walmart or Petco, they both sell it, it's a green and tan packaging, it's the only one that uses Clove-Oil...
Keep in-mind that if you actually do have a flea "infestation" in your house, and after you bath all of your animals and clean up the house and you still have fleas that are visibly hopping around, you'll need to call a professional exterminator and tell them you have birds, see what they suggest, because once you get an actual flea "infestation" in your home, it's a ***** to get rid of, and will just keep repopulating...