The ternary fission of 238-256Cf even-A isotopes is studied in the framework of the three-cluster model within the spherical approximation. The study is confined to those decays in which the accompanied light charged particle is of mass A = 10. The most probable ternary fission path is obtained as the one that has a peak in the Q-value and a minimum in the driving potential, with respect to the mass and charge asymmetries. Assuming both equatorial and collinear configurations of the different emissions of the light charged particles with A = 10, the ternary fission of 252Cf is found to be most favored with 10Be. The predicted favorable fragmentation channels of the 10Be-accompanied ternary fission of 238-256Cf, even-A, isotopes are discussed in detail. The closed-shell structure of the produced heaviest fragment is found to play a key role for the detected most favorable channels. A nucleus with a closed neutron shell or even doubly closed shell always appears as the heaviest nucleus in the favored channel of the ternary fission of all mentioned isotopes. As the isospin asymmetry of the isotope increases, the width of its cold fission valley around the symmetric mass region decreases, with more shifting towards the centre of the symmetric region. Neither the most favorable ternary emissions nor the behavior of the ternary fragmentation potential changed with the type of nuclear interaction potential (Yukawa, folding potential with Migdal force, and proximity 77). Also, almost no change is observed according to the method of emission of the fragments (equatorially or collinearly) or when the calculations are performed without inserting the nuclear interaction potential.