I think you just have to specify the boundary you need via a different mechanism. For more details see this and this. For example:
ParametricPlot3D[{v,Cos[t],Sin[t]},{t,0,2 Pi},{v,-3,3},
PlotPoints->30,Boxed->False,Axes->False,
PlotStyle->{
FaceForm[Directive[Opacity[1],Green],
Directive[Opacity[1],Lighter[Magenta,.25]]]},
RegionFunction->Function[{x,y,z},x^2+z^2>.35],
Mesh -> {{.01}},
MeshStyle->Directive[Thickness[.01],Black],
MeshFunctions->{Function[{x,y,z},x^2+z^2-.35]}]
