The mishna says that a lost object that is found by a child goes to his father. The gemara says that Shmuel and R' Yochanan disagree about who is a "katan". Shmuel holds that we are speaking of a child below bar mitzvah (Tosafos 12b), therefore the rationale for his lost object going to his father is that when he originally picks it up, he does so for his father [It is not so clear how a child is able to be zocheh in the object by picking it up, for his father. Hagahos Mordechai seems to say that only after the father actually receives it do we say that it belongs to the father. However, the Ritva explicitly writes (as implied by rashi) that the father is zocheh in it even prior to it coming into his hands. R' Yochanan disagrees about the definition of a "katan". He holds that age is not the issue. Anyone who is self sufficient is considered an adult, but the object found by any child who relies on his parent for support belongs to the father. The SM"A (270:2) quotes himself in the drisha where he explains that the rabbonon instituted this as a means of compensation and hakaras hatov for the support that they are providing without being obligated - ונראה ישר לחכמים לשלם הטוב לעושה עמו טוב, שמזין אותו אע"פ שאינו חייב לפרנסו כי אם עד ו' שנים
However, Tosafos points out that chazal only instituted this by a parent who supports a child to avoid an concern of animosity building up, but one who supports an orphan is not entitled to the objects that he finds (as paskened in the Rama 270:2). The rationale is that even if one gets annoyed and stops supporting the orphan when he realizes that he isn't entitled to his findings, we aren't so concerned because someone else is likely to feel bad for him and pick it up, which is no the case when his parents are alive.
Regarding a child who works and earns money, the Rama rules that the father is also entitled to that money due to the same takana. However, R' Akiva Eiger quotes those who disagree. The Machaneh Ephraim (zechiya 3) quotes a machlokes rashi and Ritva in Baba Metziah 92b. Rashi holds that it has the same din as the metziah which goes to the father, but the Ritvah says that chazal didn't give the father ma'aseh yadayim of his son. In regard to a daughter, the father has rights to her metziah even if she is not being supported by him (until she is a bogeres - 12 1/2). Rashi holds this is a d'oraysa, but Tosafos holds that it is a d'rabonon to avoid animosity so that he doesn't marry her off to a degenerate. Presumabely, her earnings would also be pending on the same machlokes.
Being that it is a major machlokes who is entitled to the earnings of a child, it would seem that if the child is paid directly by the employer he or she can say that they hold like the opinion who says that they can keep it. But if the employer pays the father then he can also say he holds like the opinions who say that he is entitled to her earning.