I think this is a version problem. In MMA 12.3 I get:
Show[ParametricPlot3D[R[r, s], {r, -3, 3}, {s, -3, 3},
ImageSize -> 600,
Epilog ->
Inset[Style[" 710178 ", FontSize -> 84, Red, Opacity[0.1]]]],
ParametricPlot3D[S[f[r], f[s]], {r, -3, 3}, {s, -3, 3},
PlotStyle -> Transparent]]
Update
To make the plane translucent, you could write:
Show[ParametricPlot3D[R[r, s], {r, -3, 3}, {s, -3, 3},
ImageSize -> 600, PlotStyle -> Opacity[0.3],
Epilog ->
Inset[Style[" 710178 ", FontSize -> 84, Red, Opacity[0.1]]]],
ParametricPlot3D[S[f[r], f[s]], {r, -3, 3}, {s, -3, 3},
PlotStyle -> Transparent]]