The only paints I use anywhere in my home or habitat are (human) child safe. That means they don’t contain lead, chromates or zinc, are low odor and fast drying.
These days manufacturers are using an outline of a mother holding a child’s hand to indicate a safe product.
If painting a bird’s cage it’s important to scrub it down first and remove any flaking, old paint or rust. Power-washing it and allowing it to thoroughly dry afterward is important.
Paint the cage outdoors if possible; if not, a remote and well ventilated room far away from the bird, or remove the bird completely from the home for the next 48 hours. If the paint is anything other than a quick drying, water based paint, wait 7 days.
Even a fast drying paint should cure for 48 hours and then be washed down (plain water is fine) and dried before the bird is allowed to live there.
I wouldn’t use a cage cover for at least two weeks and two more thorough washings since vapors might still be released from dry paint.
If painting inside your home – choosing a child safe paint, water based, low odor and fast drying is your best option even if the bird won’t be exposed to the actual painted surface. Look for the silhouette of mother and child on the can.
I admire you for asking this - it shows how much you care