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The question is not specific, but not entirely non-specific.

Please suggest how to compactly re-code and generalize ParametricPlot command to push in one more parameter w in Manipulate mode.

That is, how can I extend as function of function or by any other means conversion from:

ParametricPlot[{f[u, v], g[u, v]}, {u, umin, umax}, {v, vmin, vmax}]

to

ParamManipulatePlot[{f[u, v, w], g[u, v, w]}, 
  {u, umin, umax}, {v, vmin, vmax}, {w, wmin, wmax}]

Either one slider for any one of three parameters {u, v, w} should be provided for variation/perturbation or alternatively three sliders for all three parameters {u, v, w} for variation/perturbation.

In the last option I for one opine it is not about the technique of just doing it, but about demonstration of wider science of mathematical thought. Perhaps in tune with NKS philosophy, regarding multiple parameter embedments even if immersion is just in the minimum visible 2D plane.

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Manipulate[ParametricPlot[{w Sin[u + v + y], z Cos[(u + v) x]}, 
              {u, 0,  2 Pi}, {v, 0, 2 Pi}, PlotRange -> {-1, 1}],
          {{w, .5}, 0, 1}, {{z, .5}, 0, 1}, {{x, .5}, 0, 1}, {{y, Pi}, 0, 2 Pi}]

enter image description here

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