Oil is a big no mostly because of solvents. They're toxic enough that you shouldn't use them while pregnant and are a huge no for birds, even the "odorless" kind. I believe you can cut oil paint with oil for a longer drying time but reduced toxicity. I've used walnut oil for this purpose but have heard good things about linseed oil.
I have noticed some of my nicer acrylic paints smell (especially titanium white) so I tend to do research before using them - you kind of have to look up every brand/pigment, so just start with what you have.
Gouache and watercolor are fairly safe.
Obviously don't let them get into your paints. FW ink seems okay also, I've used that in the living room (which is where my tiel sleeps and where my Amazon's day cage is). It's an acrylic based ink, and acrylics don't really have fumes/odor. If there is any chance at all your bird can land in your palette, DO NOT USE TOXIC PIGMENTS. That having been said, I definitely use burnt/raw umber which contains heavy metals that are very toxic in concentrated amounts. BE SMART, don't get it on your skin if you can avoid it and for pete's sake, don't dip your paint brush into your coffee mug.
Here's a list of toxic pigments used across most artistic mediums:
From the studio: Toxicity of Pigments
I have kids AND dogs AND parrots, so I usually only paint at night, and only in mediums like FW inks or gouache/watercolor as they dry quickly and don't smell much. Don't forget to keep your paintbrushes away from your fids also - fortunately I keep mine pretty clean, but I left them out on my desk once and my toddler AND cockatiel decided to chew on them...

