Can a father and son take karate or kung-fu lessons together if part of the training involves sparring? Is there a problem with the son hitting the father?
The main question here, I believe, is whether a father has it within his authority to allow his son to strike him, which would otherwise be a terrible sin for the son.
In Y"D (240:19) the Shulhan 'Aruch writes that a father may not insist on being honored, because if the son fails, the father has effectively placed a stumbling block before him. Instead, writes the Shulhan 'Aruch:
אלא ימחול לו...שהאב שמחל על כבודו כבודו מחול.
Rather, he should waive (his honor)...since the father who has waived his honor, his honor is (effectively and as a matter of Halachah) waived.
I doubt that there is a published opinion directly on point, however, the Star-K website includes an article on a similar question. There the issue concerns whether a son, who is a dentist, may do dental work on his father in order to save the father a substantial amount of money. The article includes numerous sources which speak to the general halacha, which asserts that: (a) it is forbidden to hit or wound any Jew -- more so if you cause him to bleed, with the exception being that a doctor can cause his patient to bleed for medical purposes; and (b) there is a separate commandment that prevents a son from hitting his parent and causing the parent to bleed. (See רמ"א יו"ד סי' רמא סעי') The article cites the Chidah and others for the position that if there is another doctor available, then it is forbidden for the child to do so.
However, the Minchas Chinuch (מנחת חינוך מצוה מח), rules that if the parent gives his or her permission, then the child could even hit the parent and cause bleeding! The article cites former Israeli Chief Rabbi Yitzhok HaLevi Herzog and Rabbi Shlomo Zalman Auerbach, among others, who would follow the Minchas Chinuch in medical situations.
It is possible that the opinion of the Minchas Chinuch might open the door for a rabbinic ruling that a father and son could practice martial arts together, even if that meant that the child could cause his father to bleed. If I were to take this question to my local Orthodox rabbi, I might stipulate that both father and son will be fully padded, to make bleeding less likely. Still, local Orthodox rabbi might ask whether there is anyone else who can practice with father and son. Also, local Orthodox rabbi might very well decide that the opinion of the Minchas Chinuch is limited to medical necessity issues and that the father cannot give permission to son to hit him -- at least not hard enough to cause bleeding.
The Gemara in Sanhedrin 84b relates that several Amaroim did not let their sons to let their blood (a common cure in those days) even for the benefit of the father because if the Issur of making a father bleed.
The Rema in Shulchan Aruch, Yoreh Deah, Simun 241, sif 3 says that if there is no one else then it is permissible to do it.
Now, bruising someone on Shabbos is considered the same as blood letting because the blood leaves the main flow. In Karate, there is probably at least some bruising.
Seeing as this is not a case of Bdi-eved where no else can help because the whole Karate is unneeded, it follows that it is Ossur in accordance with the Shulchan Aruch.