I have pet birds, and it's difficult to use pesticides that won't harm them. What can I use to get rid of gnats (from decaying onions), without harming the birds? The parrots are indoors, and perpetuate the problem with fresh fruit/vegetables.

First of all get rid of the decaying onions. The gnats should go away then.

If you have gnats on house plants, soak a mosquito dunk in water overnight and water the plants with that water. The mosquito dunks carry a disease that kills both gnats and mosquitoes but doesn't harm fish, birds, or mammals.