Is this a square part with a square through-hole in the center? Exactly what orientation are you using to locate the part in the soft jaws? Something like a V shaped cut into the soft jaws to locate the part in 'diamond' orientation?
Are you facing the flip side before chamfering? If the part is not exactly level, this can have an exaggerated effect on the appearance of the chamfer widths. If the movable jaw of the vise is not exactly repeatable in position, it may lift the pocket and make it impossible to actually level the part when it is flipped and faced.
I prefer to use parallels in critical flip over applications because this makes it possible to test how the part has seated: both parallels must be tight. It is possible to use a combination of parallels with soft jaws, using the profile in the jaws to locate the part in X and Y, but mill the jaws overdepth so that two parallels will hold the part slightly above the bottom of the jaw pockets.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)