We consider the problem of the computing the boundary of the convex hull of rational parametric curves in \mathbb{R}ˆ2 and in \mathbb{R}ˆ3. The boundary of the convex hull is a semi-algebraic set and an exact representation of it breaks it down to a combination of line segments and arcs of the curve for the 2D case, and of triangles and surface patches for the 3D case. For both plane and space curves, we provide an algorithmic solution together with bit-complexity estimates. We show that the computation of the convex hull's boundary reduces in both cases to univariate and bivariate solving and to isolating roots of a univariate polynomial with coefficients in a multiple field extension. We express the bit-complexity with respect to the bit-complexity of the latter operation. Leveraging the results of Katsamaki and Rouillier [ISSAC '23], offers asymptotic upper bounds on the total bit-complexity.