The direct answer for a cockatiel
It is not safe for a cockatiel to eat food made for other kinds of pets as anything more than a rare accidental taste. A cockatiel is a small Australian parrot adapted to grass seed, greens and modest amounts of protein, and its body expects food in that pattern. Dog and cat foods are engineered for carnivores, rabbit foods for hindgut-fermenting grazers, and even other parrots' mixes are blended for larger beaks and bigger appetites. A cockatiel's curiosity makes the question practical rather than theoretical, because this species forages on the ground and tastes almost everything it finds there.
Why cockatiels run into this problem
A cockatiel out of its cage naturally drops to the floor and walks, picking at crumbs with the same behaviour it would use on fallen seed in the wild. A dog's bowl left down after breakfast sits exactly at cockatiel height, full of strong-smelling pieces the bird can carry off in its beak. Cats' wet food is softer and smellier still, and cockatiels are drawn to investigate it. The hazard is therefore built into the mixed-pet household layout, not into any failing of the bird. Managing where other pets eat and how long their food stays down matters more for a cockatiel than for a species that stays up on its stand.
What other pets' food does to a cockatiel
Kibble's fat and salt load, repeated over weeks, pushes a cockatiel towards excess weight and strains organs that handle rich food poorly, while the vitamin and mineral premix in a carnivore's diet is balanced against a dog's or cat's whole intake, not a bird's. Ground-up pieces can also be swallowed in chunks a cockatiel would never take from its own bowl. Wet foods standing at room temperature grow bacteria, and a bird that feeds face-first from a cat's dish meets those bacteria directly, as well as residues of saliva from the pet that ate there first. Ferret and reptile foods, sometimes kept in the same homes, are even richer in animal protein and further from anything a cockatiel should touch.
After a cockatiel has raided a bowl
Put the cockatiel back on its own food and water, and note roughly what and how much it took, since that shapes how closely you watch. A few pieces of dry kibble in an otherwise healthy bird usually pass without drama. Watch the droppings for a day, along with appetite, energy and breathing. A cockatiel that becomes quiet, fluffs and sits low, breathes with an open beak or tail bob, or stops eating needs an avian veterinarian promptly, and wet or spoiled food in the story lowers that threshold further. Vomiting in a cockatiel is never a wait-and-see sign.
Rules that keep the cockatiel's diet its own
Feed dogs and cats at set times in a room the cockatiel does not enter during flight time, and lift and wash their bowls when the meal ends. Sweep the feeding area, because kibble dust and dropped pieces are exactly a ground-feeding cockatiel's targets. The bird's own diet, a measured seed or pellet base with daily vegetables and limited treats, should be interesting enough that the floor holds less appeal, and foraging toys at the cage and stand give the tasting instinct a safe job. KinsLe Care Teams lets a household turn these habits into named tasks, from lifting the dog's bowl to the evening floor sweep, each marked complete with a timestamp so the routine holds even on busy days.