One observation is, that the coordinate functions can't be smooth in the corners of the convex hull, so for polynomials as coordinate functions, the only points on the curve that qualify as corners of the CH would be the end points of the curve segments.
That observation yields the sufficient condition, that the inner points of the curve segments must have non-negative barycentric coordinates with respect to the simplex defined by the curve segment's endpoints.
The Oloid and also the Sphericon however are the convex hull of two 3D curves, that are the union of an uncountable number of line-segments without being a polyhedron.
Edit
I will try to formulate some reasonable conditions for two suitable curves:
both curves must be planar, but not co-planar; this condition may however only be a sufficient condition and only cover a special case.
both curves must have finite length and contain the end-points
the containing plane of each curve must intersect the other in an inner point
if the curves are rotated into coplanar position around the intersection of the containing planes, the convex hull of the curves in coplanar orientation consists of the entire two curves and a maximum of two line-segments joining endpoints of the curves.