It’s not entirely clear what your problem is. Simply using parametric coordinates as texture coordinates should give the same result as using a cube map, at least as the vertices. However, if the triangles are sufficiently large, you’ll run into the issue that using 2D texture coordinates results in an affine mapping, causing distortion within each triangle. That can be solved by using 4D texture coordinates (u,v,0,1)/sqrt(u^2+v^2+1), which will result in the same effective texture coordinates at the vertices but a projective mapping within the triangle.
If that doesn’t give the correct result, it’s likely that your texture isn’t mapped correctly in the first place.
For reference, a cube map for a sphere with a latitude/longitude grid should look like:
