ScalingTransform[{1,1,0}]
is the project transformation to X-Y plane etc.
r[t_] := {Cos[t] (1 + Cos[3 t]), (1 + Cos[3 t]) Sin[t],
Sin[3 t]};
Show[
ParametricPlot3D[
Through@(ScalingTransform /@ {{1, 1, 1}, {1, 1, 0}, {1, 0, 1}, {0,
1, 1}})@r[t] // Evaluate, {t, 0, 2 π},
PlotStyle -> {Directive[AbsoluteThickness[3], Red],
Directive[Dashed, Green], Directive[Dashed, Cyan],
Directive[Dashed, Yellow]}, Boxed -> False, Axes -> False],
Graphics3D[{InfinitePlane[{0, 0, 0}, {{1, 0, 0}, {0, 1, 0}}],
InfinitePlane[{0, 0, 0}, {{0, 1, 0}, {0, 0, 1}}],
InfinitePlane[{0, 0, 0}, {{0, 0, 1}, {1, 0, 0}}]}]]

ParametricPlot3D
notParametric3D
. Consult the documentation. For the planes, just do three plots and set z to 0, y to 0, and x to zero and wrap in aShow
likeShow[ParametricPlot3D[...], ParametricPlot3D[...], ParametricPlot3D[...], ParametricPlot3D[...]]
$\endgroup$