Ditto to what Lucky said.
We have 7 medium & large parrots, 3 cats & 5 dogs. Everyone does very well together when we are here, but I would not trust anyone together under any circumstances if I were not here.
I even question keeping the birds out when no one is home - birds can tear through things and escape, chew on a live wire and be zapped, fly into the toilet and drown, or even fly into a window or wall and break their neck. Sorry to be so negative but birds can get into very serious trouble when their people are not watching.
Just be sure that the cat cannot learn to open the door and get into the birds. One of my cats can open doors and I am sure I would find a terrible disaster if my birds weren't safely confined when I am gone.
Best of luck.
Love Them & Lead Them,
~Elizabeth & Doug
Dog Training the Way Nature Intended