You don't need to teach a bird cognition or even the concept of cognitive speech because they already have it! Parrots have their own language and use it to communicate among themselves. There have been over 500 sounds that have been identified as words and/or phrases they use all the time so, pet parrots, been imprinted to humans, learn human language to communicate with us the same way that we would learn another language if we moved to another country and could find nobody to speak ours.
The trick is to teach them the human language the right way - something most people don't do and why, sometimes, a parrot seems to repeat something without any real context to it. But, in reality, we don't know if this is what is really happening because, for all we know, the parrot could have learned that word thinking it meant something altogether different so what sounds like nonsense to us could very well have meaning for the parrot. For example, if every time you see the parrot, the first thing that comes out of your mouth is: "Hello - Hello - Hello" the parrot might think that Hello is either his or your name (my Linus thinks that Hello is my name and that his is Hi Linus). BUT if you say "Good morning" every single morning when you first uncover his cage or see the parrot in the morning and "Good night" every single night when you either cover his cage or everybody is going to sleep, he will learn to use both salutations in context.
Having said that, parrots intelligence (like humans' and any other animal) have a range, there are below average, average, above average and geniuses and, of course, the geniuses are going to be much better at learning that the others. But even the geniuses will make mistakes when taught the wrong way. My Sweetpea Senegal is a genius, he uses cognitive language to the point that I can have actual conversations with him but, if you go strictly by the way we use language, it will seem as if he makes mistakes. For example, to him, "Eat it!" means food in general. "Peeakabird? Peekabird?" means "May I have some of what you are eating?" "Ready? Ready?" means "It's food ready?" or "Where is my food?" And, sometimes. he gets confused. For example, he likes to come with me to the canary room while I do their cages and pesters me terribly asking me questions just to get my attention on him. One morning, he was driving me crazy with his favorite "Whachudoin?" to which I am supposed to answer "I'm doing the beebee birds" (he knows the beebee birds are the canaries) when I clean the cages or "Beebee birds eat it" when I give them their food. But it was hot, I was sweating and I was getting exasperated with him and when he asked me what I was doing while I was sweeping the room, I just answered "SWEEPING!" He stopped, looked at me for a couple of seconds and then said: "What's your name?" He thought I had said "Sweetpea!" which is the right answer to "What's your name?"
And that's another one! If I ask him: "What's your name?" He will answer: "Sweetpeeeeeee!" and, if he asks me the same question, I also have to answer Sweetpea BUT I have to add: "That's my name!" (this is because I was trying to get him to learn to say that on his own but it kind of backfired because now he thinks it's a game we have). And, if I answer anything different, he will keep on asking and asking until he gets the 'right' answer to which he will reply "Right"
So, you see, it's not a matter of what they actually say but what they mean by it because, sometimes, saying the wrong thing can also be cognitive speech.