Regions for numerically defined Toroidal surfaces

Here are some modifications to ParametricPlot:

   Method -> {"BoundaryOffset" -> False}
   Mesh -> None
   PlotPoints -> {40, 20}
   MaxRecursion -> 1
   PlotRange -> All

The first one is important, since it allows the pairs of boundaries, u == 0, u == 1 and v == 0, v == 1, to match up. The Mesh option is important because it prevents the polygons being broken for the sake of a mesh line. The next two needed tweaking. Some settings resulted in meshes that had problems. Sometimes the problems were fixed with RepairMesh and sometimes not. The best chance of generating a defect-free mesh is MaxRecursion -> 0, since the recursive subdivision is not guaranteed to match up on opposite edges; usually RepairMesh fixes this problem. PlotRange -> All is safer than Automatic; I sometimes got small holes from the surface being clipped.

surfacefunc = 
  BSplineFunction[data, SplineClosed -> {True, True}, 
   SplineDegree -> 3];
plot = ParametricPlot3D[surfacefunc[u, v], {u, 0, 1}, {v, 0, 1}, 
   PlotRange -> All,
   PlotPoints -> {40, 20}, (* to get initial grid of approx. squares *)
   MaxRecursion -> 1,   (* works with 0, 1; 2 requires RepairMesh *)
   Method -> {"BoundaryOffset" -> False},
   Mesh -> None];
reg = DiscretizeGraphics[plot, PlotRange -> Automatic];
FindMeshDefects[reg]

surfacefunc = 
  BSplineFunction[data, SplineClosed -> {True, True}, 
   SplineDegree -> 3];
reg = ParametricPlot3D[surfacefunc[u, v], {u, 0, 1}, {v, 0, 1}, 
   Boxed -> False, Axes -> False];
DiscretizeGraphics[reg]

The Indexed approach produces:

reg = ParametricRegion[
    {
    Indexed[surfacefunc[θ, φ], 1],
    Indexed[surfacefunc[θ, φ], 2],
    Indexed[surfacefunc[θ, φ], 3]
    },
    {{θ, 0, 1}, {φ, 0, 1}}
];
mesh = DiscretizeRegion[reg, PerformanceGoal->"Quality"];

FindMeshDefects[mesh]