In career mode, each character can earn up to 15 Golden Parrots through their career line. Each of the five opponents give you an opportunity to win three parrots. Each opponent will award one parrot just for winning a roast, so without doing any of the other challenges, you can get a minimum of 5 per character just by winning.
No matter who you start with, the criteria for each round stays the same (although the opponents change).
Round 1: Win, 15+ pt insult, use a comeback
Round 2: Win, don't fall below 15pts, get a 2x combo (use an noun / insult two rounds in a row)
Round 3: Win, Win with 8 or less insults, 25+ pt insult
Round 4: Win, 2x combo, use 3+ comebacks
Round 5: Win, don't fall below 35pts, 35+ pt insult
If you go for perfection, this will take 5 play-throughs, 4 with full 15 Parrots, and a 5th time completing the first three.