I need to convert a function of a 2D ellipse to 3D Oblate Spheroidal Coordinate. Becauase Wolfram plot function s only works in 3D Cartesian, The resulting OS system is converted back to parametric 3D.

I need 1) Convert the equation of the 2D ellipse to the Oblate Spheroidal coordinates. 2 Plot using the ParametricPlot3D which requires to convert the elliptical equation to parametric form. But I need to create the parametric form from a conversion from the Oblate spheroidal.

I have tried rthe following

TransformedField["Cartesian" -> {{"OblateSpheroidal", a}}, 
  x^2/4 + y^2/3, {x, y, z} -> {\[Xi], \[Eta], \[Phi]}] // Simplify

This returns -\frac{1}{24} a^2 \sin ^2(\eta ) \cosh ^2(\xi ) (\cos (2 \phi )-7)

But I need the parametric form hat will allow me to plot using ParametricPlot3D.

trans = CoordinateTransformData[{{"OblateSpheroidal", 1}, 3} -> 
"Cartesian", "Mapping"]
ParametricPlot3D[{trans[{1, \[Eta], \[CurlyPhi]}]}, {\[Eta], 0, 
Pi}, {\[CurlyPhi], -\[Pi], \[Pi]}, PlotStyle -> Opacity[.5], 
ImageSize -> Medium]