Hi, I opened the example you attached and deleted two dimensions and then added two new ones and the sketch is now fully constrained. I suspect that the two horizontal dimensions didn't satisfy the polygon definition even though it seems like it would. That's why I went for a direct approach of tying the polygon center to the origin and then defining the length of the sides. You had already nailed the angle with the parallel constraint so that was all it took.
I did a new sketch similar to your example (but with the default hexagon) just to prove to myself that I could constrain a polygon and it worked fine by constraining the center, the angle and the length of the side.
Good luck!