Adding a bar legend to a 3D plot indicating surface curvature by color

Here is another way. It gives a finer scale on the bar legend.

Legended[
  ParametricPlot3D[surf[u, v], {u, -5, 5}, {v, -5, 5},
    ColorFunction ->
      Function[{x, y, z, u, v}, ColorData["Rainbow"] @ Rescale[gaucurv[u, v], rng]], 
    ColorFunctionScaling -> False,
    MeshStyle -> Opacity[0.25],
    Boxed -> False,
    Axes -> False,
    Method -> {"ShrinkWrap" -> True}],
  BarLegend[{"Rainbow", {Floor[rng[[1]], .01], Ceiling[rng[[2]], .01]}}, 9]]

Replace PlotLegends -> BarLegend[{"Rainbow", rng}] with

PlotLegends -> BarLegend[{ColorData[{"Rainbow", rng}], rng}]

or with

PlotLegends -> BarLegend[{"Rainbow", rng}, Method -> ColorFunctionScaling -> True]

to get the correct colors in the legend:

ParametricPlot3D[surf[u, v], {u, -5, 5}, {v, -5, 5}, 
 ColorFunction -> Function[{x, y, z, u, v}, 
    ColorData["Rainbow"]@Rescale[gaucurv[u, v], rng]], 
 ColorFunctionScaling -> False, 
 MeshStyle -> Opacity[0.25], 
 Axes -> False, Boxed -> False, 
 PlotLegends -> BarLegend[{ColorData[{"Rainbow", rng}], rng}]], 
 Method -> {"ShrinkWrap" -> True}]

ParametricPlot3D[surf[u, v], {u, -5, 5}, {v, -5, 5}, 
 ColorFunction -> Function[{x, y, z, u, v}, 
    ColorData["Rainbow"]@Rescale[gaucurv[u, v], rng]], 
 ColorFunctionScaling -> False, 
 MeshStyle -> Opacity[0.25], 
 Axes -> False, Boxed -> False, 
 PlotLegends -> BarLegend[{"Rainbow", rng}, ColorFunctionScaling -> True], 
 Method -> {"ShrinkWrap" -> True}]

same picture

A third approach is to move the legend outside ParametricPlot3D using Legended as in m_goldberg's answer:

Legended[ParametricPlot3D[surf[u, v], {u, -5, 5}, {v, -5, 5}, 
  ColorFunction -> Function[{x, y, z, u, v}, 
    ColorData["Rainbow"]@Rescale[gaucurv[u, v], rng]], 
  ColorFunctionScaling -> False, MeshStyle -> Opacity[0.25], 
  Axes -> False, Boxed -> False, Method -> {"ShrinkWrap" -> True}], 
 BarLegend[{"Rainbow", rng}]]

same picture

All three methods work both in Version 11.3 (Windows 10 6 bit) and Version 12.0 (Wolfram Cloud).