Rav Shmuel Vital (1598-1677) writes that the custom in his time was to permit even young children to recite Kaddish for a deceased parent, Heaven forbid. Others, however, opposed this practice. Some Poskim permitted a child to recite Kaddish only if he is at least six years old and understands the concept of praying to Hashem. There were also Poskim who forbade any child under the age of Bar-Misva from reciting Kaddish for the congregation.
Hacham Ovadia Yosef ruled that strictly speaking, if a child is at least six years old and understands the general concept of prayer, he may recite Kaddish for the congregation for a deceased parent, Heaven forbid. Preferably, however, an adult should recite the Kaddish with the child. Hacham Ovadia made an exception in the case of the Kaddish recitation following the reading of Tehillim, which to begin with is not obligatory. Since this Kaddish recitation is not an outright requirement, it may be recited by a child even if no adult recites it with him.
Hacham Bension Abba Shaul (Jerusalem, 1924-1998) ruled more leniently, allowing a youngster age six and above to recite Kaddish alone even during the prayer service. He noted the Halacha permitting a child at this age to receive an Aliya, which includes leading the recitation of Barechu. By the same token, then, a child may recite Kaddish.
Summary: A child age six or above, who understands the concept of prayer, may recite Kaddish for a deceased parent, Heaven forbid, though according to some opinions, there should preferably be an adult reciting Kaddish with him. When it comes to the Kaddish recited after the reading of Tehillim, according to all opinions a child age six or above may recite this Kaddish alone.