How to draw a surface in polar coordinates

The plots that you are trying to reproduce appear to use Plot3D rather than ParametricPlot3D

Clear["Global`*"]

g[r_, θ_] := Module[
  {t = Mod[1/r - θ/(2 π), 1]}, 4 t (1 - t)]

plt1 = With[{r = Sqrt[x^2 + y^2], θ = ArcTan[x, y]},
   Plot3D[r^2*g[r, θ], {x, -1, 1}, {y, -1, 1},
    PlotPoints -> 200,
    PlotRange -> {{-1, 1}, {-1, 1}, {0, 1.9}},
    Mesh -> None,
    Exclusions -> None,
    AxesLabel -> Automatic]];

plt2 = With[{r = Sqrt[x^2 + y^2], θ = ArcTan[x, y]},
   Plot3D[g[r, θ], {x, -1, 1}, {y, -1, 1},
    PlotPoints -> 200,
    PlotStyle -> Opacity[0.75],
    PlotRange -> {{-1, 1}, {-1, 1}, {0, 1}},
    Mesh -> None,
    Exclusions -> None,
    AxesLabel -> Automatic]];

GraphicsRow[{plt1, plt2}]