When an explosive shell travelling in a parabolic path under the effect of gravity explodes in the mid air, the centre of mass of the fragments will move

The forces leading to the explosion are internal forces. They contribute nothing to the motion of the centre of mass. The total external force, namely the force of gravity acting on the shell in the same before and after the explosion. Therefore, the centre of mass of the fragments of the shell continues along the same parabolic as it would have followed, if there were no explosion as shown in figure.