Rabbinical Commands and prohibitions:[1]
A father is Rabbinically obligated to educate his child who has reached the age of Chinuch in even Rabbinical commands. Likewise, the father must reprimand and stop his child from transgressing even Rabbinical prohibitions. However, there are certain leniencies in which the Sages do not require even a father to reprimand his child who has reached the age of Chinuch for performing a rabbinic prohibition. This matter will be explained at length in Halacha 8C-G.
For a need of the child: It is forbidden to feed a child even a rabbinic prohibition when there is no genuine need; however, when there is a true necessity for the child’s physical development, poskim dispute the matter, and in practice one may rely on the lenient view only if no gentile is available to do so, as explained in Halacha 8C.
May a child be instructed to transgress a Rabbinical prohibition for the sake of a Mitzvah: One may be lenient to instruct and allow a child to transgress a Rabbinical prohibition for the sake of a Mitzvah, as explained in Halacha 8D.
[1] Admur 343:2 “He is obligated by rabbinic law to rebuke them and prevent them from transgressing a negative commands even if it is a Rabbinical prohibition”; Admur 343:3 “This applies both to positive commandments from the Torah and to those enacted by the Sages.”; 343:6 in parentheses; 362:17; Taz 343:1; M”A 343:2; Beis Yosef 343 in name of Rav Pasi; Radbaz 1432; Olas Shabbos 343:2; Elya Raba 343:2; P”M Pesicha Koleles 2:10; Rambam Shabbos 24:11; Machalos Assuros 17:27; Nechalos 11:10; Tosefes Shabbos 343:2; Chayeh Adam 66:2; Bikurei Yaakov 657:4; Chikrei Lev O.C. 99; Y.D. 152; M”B 343:3; Kaf Hachaim 343:10, 18 and 28; Toras Refael 99; Piskeiy Teshuvos 343:1; Encyclopedia Talmudit ibid p. 192 footnote 466; p. 197 footnotes 539-543; See Toras Hakatan [Rubinovitch] Chapter 2:11-21
Other Opinions: Some Poskim rule that there is no obligation for a father to reprimand a child who is below Bar and Bas Mitzvah who is performing a mere Rabbinical prohibition. [Implication of Rama 343:1 who writes that he is only obligated to refrain the child from transgressing a Biblical prohibition; Shach in Yoreh Deah 81:26 learns that by Rabbinical commands there is no need to reprimand the child, as is the simple reading of the Rama; Rameh Mipano 111; Elya Raba 186:3 in opinion of Levush; Birkeiy Yosef 657 in name of Batei Kehuna leaves this matter in question regarding Daled Minim on other days and Chagiga 6a; Imrei Bina O.C. 14; Hagahos Chasam Sofer O.C. 343; Encyclopedia Talmudit ibid p. 192 footnote 467-473, p. 197 footnote 543]