The special syntax Line[{{0}, {1}}] is not used for 2D graphics, but for geometric regions. It means the line in 1D space, connecting the two zero-dimensional points {0} and {1}.

This is a ligher version, with only the boundary shown:

reg = RegionBoundary@
   RegionProduct[
    BoundaryDiscretizeGraphics@
     ParametricPlot[{(1 + Cos[(8 t)]) Cos[t], (1 + Cos[8 t]) Sin[
         t]}, {t, Pi/8, 2 Pi + Pi/8}], 
    DiscretizeRegion@Line[{{0}, {1}}]];
Graphics3D[{Red, EdgeForm[], Opacity[.5], MeshPrimitives[reg, 2]}]

@Gianluca Gorni Hm... can't seem to exactly find the right resources to convert the code you gave me into a MeshFunction. Perhaps when you get the chance you could help me. Thanks!

@Gianluca Gorni thank you so much! One last question (sorry!): how do I change the color of this graphic? I'm confused as to where I would put it in the coding. :)

This is so helpful, thank you all. One more question: I am trying to combine this with another ParametricPlot3D, but whenever I try the command Show[{figure1,figure2,figure3}, PlotRange->All], it returns that it cannot combine them. Any help with this part?

Here is a way:

f[t_, u_, r_] = {r (1 + Cos[(8 t)]) Cos[t], r (1 + Cos[8 t]) Sin[t], 
   u};
Show[ParametricPlot3D[f[t, u, 1], {t, 0, 2 Pi}, {u, 0, 1}, 
  Mesh -> None, PlotStyle -> Opacity[.9], PlotPoints -> {40, 2}],
 ParametricPlot3D[{f[t, 1, r], f[t, -1, r]}, {t, 0, 2 Pi}, {r, 0, 1}, 
  Mesh -> None, PlotStyle -> Opacity[.9], PlotPoints -> {40, 2}]]