Disappearing level curves on ParametricPlot3D

The mesh of level curves on ParametricPlot3D[] does not work for some functions. For many functions it works normally, e.g.:

ParametricPlot3D[{r*Cos[p],r*Sin[p],r},{r,0,2},{p,0,2Pi},MeshFunctions->{(#3&)}]


But whenever the inverse occurs, the mesh disappears

ParametricPlot3D[{r*Cos[p],r*Sin[p],1/r},{r,0,2},{p,0,2 Pi},MeshFunctions->{(#3&)}]


It seems that this issue is due to a bug (version 13.0). Do you have any idea how to work around it?

Specify the levels using the option Mesh:
ParametricPlot3D[{r*Cos[p], r*Sin[p], 1/r}, {r, 0, 1}, {p, 0, 2 Pi},