My wife thinks that our parakeets and cockatiels have some sort of bug on them which is causing her to itch.

Yes, various kinds of mites. Check out birdmite.com. People don't always succeed using the suggestions, however (I see that the site is for sale!). It can be a real problem if something is biting you. Eradication of any tiny bug that gets into mattresses and cupboards and food and rugs - anything - can be very difficult.

If your wife thinks that your birds have mites than it's time to talk to your vet. There are powders and drops available for this. Mites aren't hard to get rid of. Once the proper medication is applied just keep the bird cages clean and you shouldn't have any further problems.

Your wife's itching could also be because of allergies. Birds create a fine dust on their skin, it comes off when they groom. Many people are allergic to this dust. There's not anything that can be done about that though, other than allergy meds for your wife or re-homing the birds.