Statistical parameters of phase fluctuations of optical radiation with a cone-shape wavefront propagating in turbulent atmosphere are studied. The method of smooth perturbations is used for calculation of the variance of phase fluctuations of conical optical waves formed by conical and conoidal axicons. It is found that the variance of phase fluctuations of a conical optical wave in turbulent atmosphere is less than that of a plane wave and a fundamental Bessel beam. The dependence of the variance of phase fluctuations on the parameter characterizing the cone-shape wavefront of an optical wave, is of a two-scale character for both conical and conoidal axicons. A decrease in the variance of phase fluctuations in turbulent atmosphere is more typical for the conical axicon than for the conoidal. The spatial nonuniformity of phase fluctuations, known for a fundamental Bessel beam, occurs for conic optical waves as well; however, the quantitative value of the effect differs in this case.