Alien, the trick is to not expose the parrot to fresh organic fertilizer. As you have rightly surmised, just because something is "organic" doesn't mean it's healthy to just roll around in it. Most organic fertilizer has at least some manure-based components, and these will of course have bacteria like E. Coli in them for quite some time after application.
If you are worried about bacteria from an old application of the fertilizer, don't be. Just don't add any more to the garden if you plan on having the bird in it in the near future, perhaps a month. Most gardeners will tell you not to eat any produce from a garden within one month of fertilizer being applied. That should be plenty of time for the fertilizer to break down and decompose to a "natural" state.
This all assumes non-toxic plants in the garden, which is sort of a chuckle to me. Show me a garden that doesn't have any unwanted plants in it, and I'll give you a good deal on the Brooklyn Bridge. Who can tell if any of the weeds is toxic to parrots. If I were going to let a parrot play with pretty plants and flowers, it would probably be in hanging pots in a more controllable environment, such as a screened in patio or a greenhouse. But then again...I'm a worry-wart.