Vatican website translation:
32. The liturgy makes distinctions between persons according to their liturgical function and sacred Orders, and there are liturgical laws providing for due honors to be given to civil authorities. Apart from these instances, no special honors are to be paid in the liturgy to any private persons or classes of persons, whether in the ceremonies or by external display.
32. In Liturgia, praeter distinctionem ex munere liturgico et Ordine sacro manantem, et praeter honores ad normam legum liturgicarum auctoritatibus civilibus debitos, nulla privatarum personarum aut condicionum, sive in caerimoniis, sive in exterioribus pompis, habeatur acceptio.
Slavishly literal translation:
32. In the Liturgy, outside of the distinction maintained from liturgical office and sacred Order, and outside of the honors owed according to the norm of liturgical laws to civil authorities, let no regard for private persons or conditions, whether in ceremonies or in external show, be maintained.
The last of the norms arising from the hierarchic and communal nature of the liturgy seems to argue that the relation of the baptismal priesthood to the priesthood of the ordained be authentically signified. Thus those who assume liturgical functions OUTSIDE of ordination (e.g., members of a choir) may wear distinctive vesture, while the ordained have rather clear strictures on the vesture they are to assume as they exercise their liturgical ministries. Those who belong to the baptismal priesthood who do not assume a particular liturgical function (i.e., who pray as the congregation or assembly) could signify that by their clothing as well (“Sunday best”) but this is not as regulated.
I think the force of this norm might be to challenge practices such as distinctions in seating for the laity at a liturgy based on the ability to pay (“pew rent”) or ethnic heritage (there were cases in the United States where cultural racial segregation was carried over into the congregation’s seating at liturgy). It may also raise some questions about how groups such as the Knights of Columbus or the Boy Scouts might be properly integrated into liturgical worship.