Example
The image of the mapping (u,v) --> (1,v,u) with domain equal to the parabola, v=u^2.  (The slice of the cone z^2=xy with the plane x=1.)

s=Plot3D[{Sqrt[x  y],-Sqrt[x y]},{x,-3,3},{y,-3,3},Mesh->None,PlotStyle->{Opacity[0.85],Opacity[0.85]},BoxRatios->{1,1,1},RegionFunction->Function[{x,y},Abs[x+y]≤3],AxesLabel->{"x","y","z"}];

p=ParametricPlot3D[{1,u^2,u},{u,-2,2},PlotStyle->Thickness[0.01]];

Show[s,p]