Hmm... you have to be parallel
to something. A double-bitted axe, for example, has an axe face on two sides. I'm guessing, however, that single-bitted axes were the prehistoric norm. In that case, your sentence is saying that the face (the sharp edge) of the axe is parallel to the small part that the handle fits into. I guess I have a little trouble with this simply because the two lengths (the axe face and the whatever-it's-called where the axe handle fits) are such different lengths. A picture would certainly help, perhaps from Google Images.
Failing that, are you sure that you can't simply use "straight" rather than "parallel"?
And to throw more confusion in here, I usually think of "narrow" as coming from two sides, rather than just a single side, like a narrow waist. What about "concave" as a description of the axe head?
Finally, I have a feeling that your two-thirds is divided up top and bottom. Top third straight, middle third concave, bottom third straight. But this would give you a pretty odd-looking axe head which I doubt even primitive people would design.
You seem to be pressed for time, but I'm not sure this is an easy question to answer without a photo or drawing.
Edit: Ok, I went searching for myself and maybe I'm thinking too modern -- where a handle comes up through an opening. And I wasn't thinking stone at all.
So I found this
image. Is that more like what your axe head looks like? Where the concave portion is created so that the top of the handle wraps around it? And the
parallel two-thirds you're talking about is the
top and
bottom of the "blade" and you're not discussing the sharp edge at all? (You'll need to mentally rotate that linked image 90 degrees to the right.)
Just speculation... you'll need to clarify. Here's a link to Google Images of
prehistoric axe heads to help you choose.
