Hold-on there! Your Cockatoo isn't going anywhere until you have covered the needed Health Check by a Certified Avian Vet (CAV). Canada, like the USA have fair similar Import requirements for Parrots. A current Health Document Is Required!
Time is your biggest enemy here. Here in North America, you will quarantine in your home under the supervision of your CAV who will issue a Health Document. That document is only good for a very specific time. Once expired, the process has to be repeated...
Note, you are limited as to where you can enter Canada, once again, like the USA. With Ontario being your destination, you will be limited to Toronto, if you are lucky one or possible two additional International Import Centers in Ontario.
FYI: Highly recommend that you have your Parrot Microchipped!!! And, have that number included on all documents both Health and Transport!
So, regarding Airlines... Start with those that Provide direct flight from California to Toronto... Then contact each airline that offers that flight! From there determining each Airlines requirements. Note: They will want you to have all your paperwork completed and ready several days prior to departure and will likely direct you to Canada sources for such documents.
With a Cockatoo, (which species by the way?) buying a seat really makes sense as you can buckle the carrier in place. May need to use a belt extender!!!
NOTE: You could need 90 days to make this happen, so you have better get going.
FYI: Also, check the CITES status of your Too... If listed and your baby is under three years of age you will need a complete Hatch Certificate from the Breeder...
And, just one more Point!!! 90 days from now is in the hight of the Spring Migration. Nothing popped up last Fall, so with luck, this Spring should be the same.