Objection 1. It would seem that every parish priest can grant indulgences. For an indulgence derives its efficacy from the superabundance of the Church's merits. Now there is no congregation without some superabundance of merits. Therefore every priest, who has charge of a congregation, can grant indulgences, and, in like manner, so can every prelate.
Objection 2. Further, every prelate stands for a multitude, just as an individual stands for himself. But any individual can assign his own goods to another and thus offer satisfaction for a third person. Therefore a prelate can assign the property of the multitude subject to him, and so it seems that he can grant indulgences.
On the contrary, To excommunicate is less than to grant indulgences. But a parish priest cannot do the former. Therefore he cannot do the latter.
I answer that, Indulgences are effective, in as much as the works of satisfaction done by one person are applied to another, not only by virtue of charity, but also by the intention of the person who did them being directed in some way to the person to whom they are applied. Now a person's intention may be directed to another in three ways, specifically, generically and individually. Individually, as when one person offers satisfaction for another particular person; and thus anyone can apply his works to another. Specifically, as when a person prays for the congregation to which he belongs, for the members of his household, or for his benefactors, and directs his works of satisfaction to the same intention: in this way the superior of a congregation can apply those works to some other person, by applying the intention of those who belong to his congregation to some fixed individual. Generically, as when a person directs his works for the good of the Church in general; and thus he who presides over the whole Church can communicate those works, by applying his intention to this or that individual. And since a man is a member of a congregation, and a congregation is a part of the Church, hence the intention of private good includes the intention of the good of the congregation, and of the good of the whole Church. Therefore he who presides over the Church can communicate what belongs to an individual congregation or to an individual man: and he who presides over a congregation can communicate what belongs to an individual man, but not conversely. Yet neither the first nor the second communication is called an indulgence, but only the third; and this for two reasons. First, because, although those communications loose man from the debt of punishment in the sight of God, yet he is not freed from the obligation of fulfilling the satisfaction enjoined, to which he is bound by a commandment of the Church; whereas the third communication frees man even from this obligation. Secondly, because in one person or even in one congregation there is not such an unfailing supply of merits as to be sufficient both for the one person or congregation and for all others; and consequently the individual is not freed from the entire debt of punishment unless satisfaction is offered for him individually, to the very amount that he owes. On the other hand, in the whole Church there is an unfailing supply of merits, chiefly on account of the merit of Christ. Consequently he alone who is at the head of the Church can grant indulgences. Since, however, the Church is the congregation of the faithful, and since a congregation of men is of two kinds, the domestic, composed of members of the same family, and the civil, composed of members of the same nationality, the Church is like to a civil congregation, for the people themselves are called the Church; while the various assemblies, or parishes of one diocese are likened to a congregation in the various families and services. Hence a bishop alone is properly called a prelate of the Church, wherefore he alone, like a bridegroom, receives the ring of the Church. Consequently full power in the dispensation of the sacraments, and jurisdiction in the public tribunal, belong to him alone as the public person, but to others by delegation from him. Those priests who have charge of the people are not prelates strictly speaking, but assistants, hence, in consecrating priests the bishop says: "The more fragile we are, the more we need these assistants": and for this reason they do not dispense all the sacraments. Hence parish priests, or abbots or other like prelates cannot grant indulgences.
This suffices for the Replies to the Objections.
Source: Summa Theologica (New Advent)