If the non-Uniform loading is of the type of parabola then for calculating the radius of gyration?

The net force will act at the centroid of the parabola

The net force will act at the centroid of the parabola. Whether it be a parabola or the cubic curve the centroid is the only point at which the net force act. Force can’t be acted horizontally if the loading is vertical. Hence whatever be the shape of the loading, the centroid is the point of action of net force. Thus the use of centroid.