What you will need:
Adjustable sock loom (adjusted to any size)
2-5 different colors of yarn (body, beak, mouth, white[eye], black[eye])
BODY - Begin just like a sock.
BEAK - When you get to the heel, switch to another color (probably orange or yellow) do it exactly as you would, but decreasing to ONE peg, then back out. [OPTIONAL: Right after you get down to one peg, you can switch to another color for the inside of its mouth. Probably pink.] Knit a few rows around. Do another heel in the exact place of the first and in the exact same way. [OPTIONAL cont'd: Do the pink when decreasing, and switch to the yellow or orange when increasing.]
HEAD - Switch to the first color of yarn. Continue knitting and finish like you would a sock (and DO tie off).
EYES - Using a yarn needle, for the white of the eyes, sew an asterisk shape, or a star-like shape. Do this on both eyes. Then sew a much smaller asterisk of black in the middles of the white for the pupils.
WINGS - Adjust the loom so that it is 2 (technically 4) pegs closer to the stationary short edge. Do (only) a heel out of the body color. Again, decrease to one peg, then increase. Tie off. Do another of these. With a yarn needle, sew them closed (separately) and then sew them to the body of the puppet in their appropriate place.
I admit, I have not tried this, I only thought of it because the heel on my sock that I am making turned out like a beak. I have no idea if all of this would work right. If you know that it would not work right, please tell me in the comments. Or even if you have a concern, please tell me. Or especially if you have tried this! Unless someone confirms that this method would work, and unless you have lots of yarn, I suggest not trying it because it probably wouldn't work... Sorry! I feel kinda guilty, like I am giving you wrong info... I am not! It is just an idea!!! ;D