Augustine notes that the term is also connected with religion; in passing in the City of God (10.1) he says:
"Piety," again, or, as the Greeks say, εὐσέβεια, is commonly understood as the proper designation of the worship of God. Yet this word also is used of dutifulness to parents. The common people, too, use it of works of charity, which, I suppose, arises from the circumstance that God enjoins the performance of such works, and declares that He is pleased with them instead of, or in preference to sacrifices. From this usage it has also come to pass that God Himself is called pious, in which sense the Greeks never use εὐσεβεῖν, though εὐσέβεια is applied to works of charity by their common people also.
This passage, which is not primarily concerned with pietas itself but with the right word to use for worship of God, can be read more than one way; I think the proper way is to read it not as saying that 'worship of God' is the primary meaning of pietas but rather the opposite: the usual word used by people to describe the worship of God was 'pietas'. But, as he notes, the word also covers dutifulness to parents and merciful works (our English word 'pity' is from pietas used to cover merciful works). It is possible, however, that Augustine is rather reporting, incidentally, a sign of how Christianity introduced a new understanding in the old term, relating it to caritas or love. John Calvin, for whom pietas is a very important concept, carries forward the Augustinian account in a number of places, including his Commentary on Seneca. He reads Augustine as taking pietas primarily to mean worship of God as Father, and takes the devotion to parents as being a spillover from this that even the pagans recognized, namely, that worship of God requires honoring one's parents.
When Aquinas discusses pietas in the Summa, he is looking at both Cicero and Augustine, and to accommodate both he develops a position of some nuance. If we are talking about virtues (ST 2-2.101), the virtue of pietas is concerned with devotion to parents; by a sort of extension we can talk about worship of God as piety, but the proper name for the virtue concerned with worship of God is religio. However, there is another kind of thing that is called pietas in Christian theology (ST 2-2.121), and that is the gift of the Holy Spirit. This piety is a special divinely given capacity to devote oneself to God as Father, and through this, to the saints as kindred and heaven as patria. Thus it retains an analogy with the virtue of piety, but is distinct from it. The virtue of piety in Aquinas is understood in a highly Ciceronian way, allowing for the modifications of Aquinas's general account of virtues. Piety is devotion to one's parents, and, through one's parents, to one's kindred and patria. It is a "potential part" of justice, that is, it is not justice in a strict sense but it is justice in a broad sense; the reason is that it does not involve a "legal debt", in which one owes something that can be precisely and definitively paid back, but a "moral debt", in which we owe something that cannot be repaid to a point at which we can definitively say that we are now equalized and no longer in debt. This is because our parents are the origin of our being and our government, and have birthed and raised us. This debt requires two things of us: duty, which is service and support, and homage, which is respect and remembrance. The service and support is not absolute; one of the interesting features of his account is that parents owe their children ongoing support to the extent of providing for their children even after death, whereas children owe their parents ongoing respect but only sporadic support as necessity requires. By a sort of spillover, piety toward parents involves respect and service to kindred generally, and to all fellow-citizens, and to all friends of our patria, not all equally, but in gradation. Aquinas claim that the Ten Commandments specifically command filial behavior rather than civic behavior for precisely this reason: honoring parents takes precedence over, and is the source of, honoring one's patria.
One of the things Matteo Ricci, a Jesuit who certainly knew his Aquinas, was very impressed with when he went to China was the Chinese pietas, which he regarded as showing that they had in many ways a very clear understanding of natural law, and on which point of natural law he regarded them as unexcelled by any other nation in the world, at least as far as external practice went. And the pietas he was noting was precisely the Chinese devotion to parents, what the Chinese call xiao. Identifying virtues across long-separated cultures is tricky business, but it would be foolish to suggest that it cannot be done, since difference of cultures governs the expression of human nature in virtue and vice rather than human nature itself. And there is no doubt that Chinese xiao is Latin pietas, and the early recognition of this is why the word xiao is normally translated as "filial piety". I notice that more recent translators often try to get away from that phrase because of the English connotations of 'piety'; this is unquestionably a result of the fading of classical studies, because anyone with a decent knowledge of Latin would see the point of the older translation at once. It's an interesting feature of Modern English that it has no real equivalent, except the older phrase 'filial piety', which is itself hardly understood anymore. No translation of xiao into Modern English will ever be as good as the older Latin translations of it as pietas, from which we get the English 'filial piety' translation. I don't think that there can be any question that the Romans would have recognized Chinese xiao as pietas, even if they thought that some of the practices were strange, nor that the Chinese would have recognized that in talking about pietas, the Romans were talking about xiao. Both cultures were Imperial cultures based on earlier cultures in which family was extraordinarily important, and these Imperial cultures could only justify themselves in familial terms, and both continued to see devotion to parents as one of the most important foundations of civilized life. And the accounts of how it does these things -- ground civilized life and govern Imperial culture -- are actually quite similar. There are undeniably differences, due to differences in familial circumstances, historical accidents like differences in what the philosophers and politicians specifically discuss or Roman interaction with Greek culture, and the like. Aquinas might have managed to convince some Romans that active parental support of children was a more important thing than active filial support of parents, but he probably would not have convinced the Chinese. But the differences are fewer and less significant than one might expect. Matteo Ricci was quite right on this score.
Thus, if we are looking for a major classical treatise on which to ground future discussion of pietas, we could hardly do better than look at the Xiao jing, the Classic of Filial Piety, an anonymous work from the fourth or fifth century BC. Traditionally the content is attributed to Confucius, but very indirectly; it depicts a conversation between Confucius and his student Zeng Zi, and was commonly thought to have been collected together by Zeng Zi's students out of things Zeng Zi had told them about the conversation. In it Confucius argues that xiao is the source of virtue. We receive our living bodies from our parents, and because of this xiao starts with refusing to do anything that harms them. This spills over because one way to harm one's parents is to harm their reputation. Therefore one should strive to act in such a way that one's deeds will glorify one's parents through the ages. It therefore includes taking care of oneself, doing great and virtuous things in general, and acting appropriately, as well as direct support and respect for the parents. On the basis of this the Emperor, out of xiao, will act as a model Emperor, thus encouraging the virtue of his people; likewise, the princes will act as model princes, the ministers as model ministers, the officials as model officials. Out of xiao, the common people, out of concern to support their parents, will also act prudently, learning to plan ahead and to make good decisions. This constitutes the main argument of the text; the rest can be seen as a sort of commentary on this idea. The man of xiao will respect his parents, work to support them, and display the appropriate public emotions on when they are sick or when they die. He will avoid arrogance to those under him, quarrelsomeness with those equal to him, and insubordination to those above him. Xiao does not involve mindless devotion, however, since, just as a good minister will remonstrate with the Emperor if the Emperor is doing something wrong, so the good son will advise his parents to help them live a virtuous life.