Halacha forbids causing a bloody wound on Shabbat. Therefore, when a parents hits a child, in addition to the psychological problems that often result from this disciplinary technique, if this occurs on Shabbat they transgress a Torah violation, as well. It should also be noted that a parent who hits even a young child from the age of three or four violates the prohibition of "Lifnei Iver" (causing others to sin), because the child will likely strike back at the parent. Thus, by hitting a child one causes him to commit a sin, and, on Shabbat, inflicting a bloody wound also constitutes a Shabbat violation. In general, hitting children should be discouraged.
Although the Torah prohibition relates only to causing bleeding on Shabbat, it is forbidden by force of Rabbinic enactment to cause an injury whereby blood collects underneath the skin, even if it does not leave the skin. Occasionally people frivolously approach their friends and pinch them, which leaves a black-and-blue mark on the skin. Beyond the general problem of childish, frivolous behavior, doing this on Shabbat also entails a violation of Halacha.
Summary: Hitting children should be discouraged as a disciplinary technique, and on Shabbat, it constitutes a Torah violation if it results in bleeding. Causing a bloody wound on Shabbat violates a Torah prohibition, and causing a wound whereby blood collects underneath the skin is forbidden by force of Rabbinic enactment.
