I have 6 strictly indoor cats and, right now, 8 dogs because I've been caring for a friend's dog for the last month. The cats are all put in a room in the morning and stay there during the day when the birds are out and let out in the evening when the birds are back in their cages. There is no interaction, closely monitoring or not, between my birds and my cats. Most of them are old (21, 17, two are 15) and 2 are under a year old and, although they are well fed and 'good' cats, I would not risk their reacting to a bird flying over their heads with a reflex swap of their paw. No person is quick enough to prevent this.
Leave the radio on and lots of chewing material. Some people leave the TV on for them but I don't like to expose mine to the pulsating light of the TV as it causes our brains to produce alpha waves (I don't know if it does the same to parrots but just in case...)