In North America those very tiny Ants are referred to as 'sugar' Ants. This since they appear like from nowhere when food, especially sweets (like fruits) show up.
To slow and to a point Stop them you have to start from two sides. First crazy cleaning of any area where the 'good stuff' sits and you have seen them. Use lots of bowls and tight lids. Now from the other side. Long slow walk around the outside of the house. Pull /cut back any bushes, flowers, plants in general including grass away from the foundation of the home. Target any place where the ground (soil) makes contact with the structures' wood surfaces and pull it back and away from the wood. Sweep porches, stairs and door openings of any build-up of sand /soil. With the windows and doors closed, use a industrial strength Ant Killer full around the home taking care to using it with care in those areas that foot traffic will carry it into the house. Example: Spray on the lift (back) of the step and not on the step itself. The Ant traps are great indoors, but take great care to pick them up before you let your Parrots out!
There are Sprays that can be used inside the home, but it would be safer to take the Parrots out of the home for the day. Example: Load the Parrots up for a ride, selectively spray areas that Parrot rarely go i.e. inside cabinets, etc... Never spray the Parrots cage or anything else the Parrots common are, like the floor.
Best of luck! It will take a little crazy time, but you can greatly reduce and possibly eliminate them.
