Is it permissible for a father to make his children give him a massage?
What we understand from the question is that enjoyment that the father feels from having these parts of the body massaged is of a normal type of enjoyment, such as feeling relaxed, but it is not accompanied by any kind of pleasure or sexual desire towards the one who is giving the massage. If that is the case, then there is nothing wrong with him being massaged by his sons and daughters on the parts of the body mentioned, whether they have reached puberty or not.
But if they have reached puberty or have reached an age where they may be desirable, then the father must cover the area between the navel and the knee before them; they may look at other areas apart from that.
Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (7/80): As for a man with another man, each of them may look at whatever is not ‘awrah of the other. Two views have been narrated concerning its definition, one of which is that it is the area between the navel and the knee, and the other is that it is the private parts. End quote.
With regard to women, the scholars have explained what it is permissible for her to see of her mahrams. The great scholar al-Nafraawi al-Maaliki said in al-Fawaakih al-Dawaani (1/287), when speaking of washing the deceased: If one of his female mahrams is present, she may wash him, and his ‘awrah should be covered from the navel to the knee, because she may see of him what another man may see, which is everything except the area between the navel and the knee. End quote.
This applies if they have reached puberty as we said, or if they have reached an age where they may be desirable. But if they are younger than that, the matter is simpler.
All of this applies if the matter is as we have understood from your question, and that what is meant by enjoying is natural physical relaxation.
But if what you mean is pleasure and enjoyment of the one who is giving the massage, then it is haraam and is a door to greater evil. In that case what is required is to prevent this evil.
But we do not regard this as something recommended even in the permissible form, especially if he is old and if it is done regularly, because that may lead to a loss of respect between the children and their father, and so as to block the way to any feared negative consequences, especially in the case of those who live in the west, and you cannot be certain what the children may be picking up from the society around them with regard to such matters.
And Allaah knows best.