[A] man's children are more lovable to him than his father, as the Philosopher states (Ethic. viii). First, because parents love their children as being part of themselves, whereas the father is not part of his son, so that the love of a father for his children, is more like a man's love for himself. Secondly, because parents know better that so and so is their child than vice versa. Thirdly, because children are nearer to their parents, as being part of them, than their parents are to them to whom they stand in the relation of a principle. Fourthly, because parents have loved longer, for the father begins to love his child at once, whereas the child begins to love his father after a lapse of time; and the longer love lasts, the stronger it is, according to Sirach 9:14: "Forsake not an old friend, for the new will not be like to him."(Aquinas ST 2-2.26.9)
I've always thought the first and third are especially interesting arguments, because I think there is almost certainly something to the basic idea, but it's very difficult to pin down exactly how to understand it in a way that goes beyond a vague sense of analogy. (It's also a good example of a way in which mereology can arise in unexpected places.) The basic idea, of course, is that parent is to child as whole is to part. Aquinas, of course, is not claiming that there aren't defective parent-child relationships, in which it does not turn out this way; the point, which has some precedent in Aristotle but is taken well beyond anything in Aristotle himself, is that parenthood as such involves taking one's children to be in some way a part of one's own self, while the reverse relationship of being a son and a daughter involves taking one's parents to be in some way a whole of which one's self is a part. This is not intended to be a metaphor. And it does seem that there is some broad, morally relevant sense of the term 'self' where both of these end up being true. The basic norm for parenthood in Aquinas's account is necessarily Christian love of neighbor -- loving others as oneself. Thus the parent is to love the child as himself or herself, and the child is to love the parent as himself or herself. But this does not mean that they are symmetrical, because it is modulated by this part-whole asymmetry. The parent ideally loves the child as himself or herself, but in a sense the child is already a definite part of the parent's self. The child ideally loves the parent as himself or herself, but for the child the parent is much more like an environment, a background, or, to use the mereological terms, a whole of which the child is a part.
In the Sentences commentary (In III Sent d 29, a 7), Aquinas links this idea to the causal relation between parent and child. For the parent, the generation of child is not really all that much different than the development of a body-part; the child is a res patris (a thing belonging to the parent, or, sometimes more narrowly, a thing of the father)and res patris diligentis est, ut membrum ipsius (the thing belonging to the loving father is as his own body-part). The child is in some sense a natural expression of the parent, whereas the parent is more like the world in which the child finds itself. Because of this, Aquinas argues, as he does above that there's a legitimate sense in which the love of the parent for the child is capable of greater intimacy (nearness) than the love of the child for the parent.
The mereological asymmetry does other work in Aquinas, although it's not often brought out in a way to make it blindingly obvious -- for instance, it is the reason that parents have authority over children, namely, that as the 'whole' they are natural caretakers for the common good they share with the child, which takes precedence over the individual good of either the parent or the child -- but we really don't need to get into it for the basic idea. (Although it's worth noting how often Aquinas puts social order in mereological terms.) As I said, it's one of those ideas that seems to have something in it -- even if it turns out unworkable as is, it seems to capture something -- but it's difficult to get traction on this idea beyond the basic idea itself.
