Consider a very simple vector field in cartesian coordinates: {x1,x2,x3}. Using the Mathematica function TransformedField, transform it to parabolic cylindrical coordinate system.That looks

transvectorfield=TransformedField[ "Cartesian" -> "ParabolicCylindrical", {x1,x2,x3}, {x1,x2,x3} -> {u1,u2,u3}]

Now with the vector field obtained apply the "inverse" transformation; i.e., go now from parabolic cylindrical coordinates to cartesian coordinates. That looks like

TransformedField["ParabolicCylindrical" -> "Cartesian", transvectorfield, {u1, u2,u3} -> x1, x2, x3}]

Now, the problem is that you do not recover the original field. If the same is done from cartesian to spherical and back, it works fine. Also works okay between cartesian and cylindrical, but is wrong in this case. The same mistake is found if you go from cartesian to elliptical cylindrical and try to go back.