Tucked away and easy to miss in Aquinas's many discussions of virtue is the virtue of observantia (2-2.102), closely related to pietas and, like pietas, a potential part or auxiliary virtue of justice. We might indeed, given things that Aquinas says about it, regard observantia as a potential part of pietas.
Pietas is giving direct and indirect honor to our parents, to whom we owe a moral debt; that is, we honor our parents directly for bringing us into the world, and we honor them indirectly by giving honor to their family and to their patria or homeland. But there are ways in which we may be related to others who are not our parents that are in some way or other analogous to the way we are related to our parents. In one case, that of God, the relation is more fundamental and complete than that of our relation to our parents, and this concerns the virtue of religio; in other cases, the relation is looser, and it is these that are the concern of observantia, which is the virtue of giving proper honor to those in positions of dignity or excellence. Like many of Aquinas's basic definitions of potential parts of justice, the definition is derived from Cicero.
In order to develop this idea, Aquinas uses the notion of pater; since observantia is analogy-based, nothing much turns on whether we read it in its specific meaning of 'father' or its generic meaning of 'parent', but the same word in relation to piety has to be understood as including, not excluding, our mothers, so it is certainly more appropriate here to think of it in terms of 'parent'. A parent is a principle of generation, upbringing, and learning, and of whatever is relevant to the completion of a human life (pater est principium et generationis et educationis et disciplinae, et omnium quae ad perfectionem humanae vitae pertinent). Just as a parent shares in a particular way the nature of a principle, which God has in a complete way, so too certain other people share in a particular way the nature that a parent has in a complete way. We see this in our tendency to extend respect given to parents to other people under limited conditions, because they have a similar sort of charge, but over a narrow field. Aquinas specifically mentions the prince of a city (for civil matters), the dux or general (for military matters), and the teacher (for learning). But there are many, very many, more ways in which one may involve something like this notion of a parent.
Aquinas identifies two major acts associated with observantia: honor and worship (cultus). Honor is recognition of an excellence; worship is deference in which one obeys someone worthy of honor or repays benefits received from them in some appropriate way. Worship in this sense thus adds to honor the notion of decent action between human beings. But there are a lots of different kinds of things we call 'honor' and 'worship', not all of which are relevant to observantia itself. Thus, for instance, if you are a soldier giving obedience and showing respect to your general, you have a well-defined obligation of obedience. This makes that kind of honor a matter of justice in the strict sense, not of a virtue like observantia that deals with less-defined matters and can only be regarded as justice in a broad sense. Thus the kinds of honor and worship that are relevant to observantia arise when there is not any precise action that we have to do given our position in society. Thus one might show respect to a teacher even though one is not that teacher's student, because the role of 'teacher' is itself something that has an excellence with respect to the common good of society. Likewise, one might show respect to a mother even though you are not related to her in any determinable way at all, because motherhood is itself an excellence, given its importance to the healthy functioning of society. And, of course, since there is no precise obligation, one would tend to render such respect on a sliding scale, with those who fulfill these roles with the greatest and clearest virtue being given the greatest honor. If anyone were to ask why we do this given that we don't strictly have to do so, we would say that we do it because, despite not being strictly required by our position, it is nonetheless a good thing to do.
One of the things that makes observantia interesting as a virtue is that it is a particularly promising locus for comparison between Western philosophy and Chinese philosophy, since it has many analogies to the Confucian conception of good social relations. Like the Confucian idea, observantia in society at large is derivative from pietas toward our parents, which serves as a kind of foundation for it.