We don't have enough information to give a solid answer. In the future, let us know more about the artwork, specifically, what kind of file it is. PSD? JPG? AI? EPS?, etc?
If it is AI or EPS, is it an embedded image (or images) or a bunch of paths and shapes? (Any other format, you would be better off asking in another category of the forums, unless you are asking how to redraw the whole thing.)
If you have the image in AI, here are some possibilities:
1) The stems will be easy if the artwork is vector already. Just select them and set your fill color to what you want. (It might already be white, btw.) If the artwork is NOT vector do as in #3 below. Use the optionals sparingly here, unless the artist in you has a strong feeling about it.
2) You are in a bit of luck with the heads - they are all just scaled/rotated/reflected copies; you only need to "fix" one, then copy it (or just the fix) and correct its orientation.
3) To fix the 1, I'd use the pen or curvature tool to draw a path around the biggest head. Fill the path with the desired color, set the stroke to none, move it behind the existing artwork in the layers panel. Optional #1: Don't trace the outline too closely - leave some variance to the inside and out. Gives some visual variety. Optional #2: Feather the path. A lot or a little to your taste. Possible Problem #1: If the logo/image is not vector it is likely a black and white image. (What I see from the Cow is, anyway.) You will need to go into Photoshop or another image editor and remove ALL the white - even expand the selection just a bit to bite into the black just a bit to remove most of the gray edges along the black.
As in most of my responses, this morphed as I wrote it. :) It should solve most of your problems with this logo if you have the skills in AI & PS.