First, you get a parrot deed. Don't pay too much for them. Seriously. They are not that uncommon.
(Edit: If you want to get your own, they drop from creatures like swoops eagles near the Spirituality Shrine on the way to Twisted Weald near the cu sidhes.)
Second, your parrot needs a home, so buy or make a parrot perch. This is a carpentry item. It -CAN- be furniture dyed any color you want, so I suggest you save your money and buy a regular wood one--unless you want it to glow that special blue of bamboo wood or the pure white of petrified.
Third, find a nice place in your house to put your perch down. I highly suggest an open area like a balcony or patio because it can poke up through the floor above sometimes. (If it pokes through the ceiling above it, you have to go upstairs to feed it wafers. I have mine on my balcony.) Do NOT lock down your perch yet.
Now, take your parrot deed and click it and then click the perch bar and your parrot will appear. Don't worry if it's backwards or sideways, it turns all around on its own while it flaps its wings and talks. Now that your parrot is placed, you MUST lock down your perch.
If your perch decays from not being locked down OR you axe your perch, your parrot will vanish completely.
______
Moving your parrot:
To move your parrot, simply double click your parrot and it will appear as a deed in your backpack. It will retain any levels it has gained while under your care.
If you are going to change anything in your customized house, pick up your parrot FIRST by double clicking it, or it will warp off its perch, end up across your house, and you may not be able to pick it up. Remember, when you put your parrot back, unlock the perch, place the parrot, lock the perch.
_____
Leveling your parrot:
The leveling process of the parrot is simple. It needs Time and Food.
Feed it wafers which you buy at 1400 gold a pop at the Trinsic gold stone. The wafers do not stack, which scatters them in your pack, but you won't accidentally feed it more than one at a time, which is good.
Between feedings, I have had the most success by waiting a day or two. I can get my parrot to level usually in 10 wafers or less, if I wait a day or two between feedings.
The max level of your parrot is 20. The higher your parrot is the more your parrot will say and the more bawdy he becomes!
I don't want to 'spoiler' what he says at each level, but I will say it's fairly mundane until you get to around 7 or so. And level 10+ is naughty parrot! 
Note: Your parrot will not die or get mad and fly away if you do not feed it.
So, feeding is optional, you only have to do it if you want your parrot to level, and you don't have to do it at all after your parrot is level 20.
