How to obtain the boundary of a two-dimensional ParametricRegion surface in order to apply Stokes' theorem to compute surface integrals?

Question

On the help page of the SurfaceIntegrate, an example of Stokes' theorem is listed:

Compute the Curl g of a vector field f:

Clear["Global`*"];

f = {-y + x  y^2, x, z};

g = Curl[f, {x, y, z}];

The surface integral of g over an open surface is:

reg = ParametricRegion[{Cos[t]  Cos[p], Cos[t]  Sin[p], 
    Sin[t]}, {{p, 0, 2  Pi}, {t, 0, Pi/2}}];
SurfaceIntegrate[g, {x, y, z} \[Element] reg]

(*2 Pi*)

This is the same as the line integral of f over the boundary of the surface:

bound = ParametricRegion[{Cos[t], Sin[t], 0}, {{t, 0, 2  Pi}}];
LineIntegrate[f, {x, y, z} \[Element] bound]

(*2 Pi*)

Here is the ParametricRegion form of the bound directly provided.

My question is, if the ParametricRegion form of the boundary is not directly given, how can one obtain the boundary ParametricRegion form from a two-dimensional ParametricRegion surface?

It would be best if the boundary is also in the form of ParametricRegion, so it can be directly used for LineIntegrate.