Use Convert to Nurbs tool to convert the sweep object into a nurbs object.
Note, parametric mapping works on NURBS objects only.
Use Reconstruct Nurbs tool to create a uniform distribution of nurbs control points on surface.
Create a Material and drag it onto your nurbs object.
Use Edit Texture tool to control size, location, and orientation of image map on object. Be sure the Parametric option is selected in Tool Options. Observe the Horizontal and Vertical Tiling size is a percentage of the surface instead of an actual distance.