A common question at traditional Latin Mass parishes and chapels this time of year is: Who is allowed to chant the Passion?
The answer: Three deacons, or priests vested as deacons.
There are no exceptions to this rule outside of the schola being permitted to chant the full crowd's parts. We are talking about the Gospel of the Mass. The Passion of Saint Matthew is the Gospel for Palm Sunday. The Passion of Saint Mark is the Gospel for Tuesday in Holy Week. The Passion of Saint Luke is the Gospel for Wednesday in Holy Week. The Passion of Saint John is the Gospel on Good Friday.
According to Sancta Missa:
Q. On Palm Sunday and Good Friday, during the singing of the Passion Gospel, are laymen permitted to sing the role of the Synogaga or Chronista?
A. The Rites of Holy Week by Fr Frederick McManus, published in 1956 by the St Anthony Guild Press, Paterson, NJ, states that, the "Passion is divided into 3 parts: the narration (C for Chronista), the words of Christ (+), and the words of the crowd or of anyone else (S for Synogoga)...The choir (even of lay persons) may take the part of the crowd; the 3rd deacon then sings only the words of individuals (i.e. Pilate, Peter, etc)." (page 69).
McManus states that "the restored order of Holy Week does not provide for the case in which 3 deacons are not available to chant the Passion. It would seem that then the deacon of the Mass should read or chant the Passion in a clear voice, since the entire Passion now constitutes the Gospel of the Mass."In the Ritus Simplex the priest singing the Liturgy (i.e. Missa Cantata), has the option of just reading the Passion Gospel on Palm Sunday and on Good Friday instead of singing it. If he chooses to sing it, he must sing all three roles, but if a choir is available they could sing the parts of the crowd (Synagoga) that are not the voices of individuals (i.e. Peter, etc)
Summary: help your local clergy learn the chanted parts of the Passion, starting with the two most often sung (Matthew and John).
Musica Sacra has each part here, here and here:
In the meantime, laymen chanting the Passion on Palm Sunday or Good Friday is as illicit as laymen chanting the Gospel at Mass any other day of the year.