Theoretical investigations are carried out of optical transitions of a polar disc with a conical disclination and under the influence of a parallel applied uniform magnetic field. Additional confinement of the electron is due to an intrinsic electric confining potential of the polar disc modelled by any of the forms: infinite polar square well (IPSW), parabolic potential (PP) and shifted parabolic potential (SPP). As is well known, the parallel applied magnetic field lifts the double degeneracy of the non-zero azimuthal quantum number m electronic states. This Zeeman splitting is such that the ��>0electron energy sub-bands increase monotonically with an increase of the magnetic field, while the ��<0states initially decrease as the magnetic field is increased. Now, in systems with cylindrical symmetry, the allowed optical transitions are those between the electron’s states whose azimuthal quantum numbers differ by unity. The conical disclination is characterized by a kink parameter which is ��<1 for a segment cut off from the disc and ��>1 for a segment introduced into the polar disc. An increase of |��| leads to a decrease of transition energies, which in turn gives rise to an increase of the corresponding transition rates of optical transitions. Thus, peaks of transition rates get red shifted as the kink parameter increases. Additionally, the magnitude of the transition rates increases with the increasing value of the kink parameter. The magnetic field enhances transition energies involving states with angular momentum in one direction (here, those with positive angular momentum number m), while it decreases those involving states with angular momentum in the opposite direction (negative m states). It has also been found that parallel magnetic field blue shifts peaks of rates of transitions involving the ��>0 states, while it red shifts peaks of those involving the ��<0 states. The parabolic potential enhances transition energies, while the shifted parabolic potential reduces the transition energies. Consequently, the parabolic potential blue shifts peaks of transition rates, while the shifted parabolic potential red shifts the peaks. The results presented here suggest that a conical disclination and the overall confinement potential can be employed to tune and modulate the optical transition rates of a quantum disc.