A new framework for the parametric reconstruction of curved fibres from glass fibre-reinforced composite X-ray computed tomography data is proposed. It allows us to detect fibres in a fibre-reinforced polymer sample from a low-dose, low resolution computed tomography scan. An efficient curve representation is then used for each detected fibre, of which the parameters are estimated directly from few 2D high-resolution projection images. The framework is validated on both simulated and real data of glass fibre-reinforced polymers. The generated results demonstrate that it is robust to noise and requires less than 10 high-resolution projections to obtain reasonable fibre estimates. The method can also improve upon existing estimation frameworks relying on full 3D scans.