Well, it depends. What's the existing context? Are all the parish Masses celebrated with the same music?

If we're thinking of an early Sunday Mass that is normally recited with no music, then adding a Marian hymn doesn't remove something else (except the silence).

If the principal Mass of the parish is already celebrated in a more complete way, with proper chants at the entrance, offertory, and communion, then it wouldn't be very fitting to bump the prescribed offertory chant-text of the day in favor of another hymn.

I think of the famous example of Pope Leo XIII who added the St Michael prayer at the end of Mass: but only after recited "Low" Masses.

Incidentally, is the rosary said before the Mass? If so, perhaps a good time for a Marian hymn would be at the end of that rosary.

I have heard complaints that a Marian hymn is not suitable at communion, but this is untrue, the Magnificat is explicity authorised by the Graduale Simplex p462 (with the antiphon My soul glorifies His holy name).

Agreed chonak, the Magnificat is Praise of God by Mary. Our praise of Mary would be wholly inappropriate. (I was just about to add that, recalling As I kneel before you).

Where she is queen, He is King.

The Vatican II document Lumen Gentium ends like this:

68. In the interim just as the Mother of Jesus, glorified in body and soul in heaven, is the image and beginning of the Church as it is to be perfected is the world to come, so too does she shine forth on earth, until the day of the Lord shall come,(304) as a sign of sure hope and solace to the people of God during its sojourn on earth.

69. It gives great joy and comfort to this holy and general Synod that even among the separated brethren there are some who give due honor to the Mother of our Lord and Saviour, especially among the Orientals, who with devout mind and fervent impulse give honor to the Mother of God, ever virgin.(24*) The entire body of the faithful pours forth instant supplications to the Mother of God and Mother of men that she, who aided the beginnings of the Church by her prayers, may now, exalted as she is above all the angels and saints, intercede before her Son in the fellowship of all the saints, until all families of people, whether they are honored with the title of Christian or whether they still do not know the Saviour, may be happily gathered together in peace and harmony into one people of God, for the glory of the Most Holy and Undivided Trinity.

Each and all these items which are set forth in this dogmatic Constitution have met with the approval of the Council Fathers. And We by the apostolic power given Us by Christ together with the Venerable Fathers in the Holy Spirit, approve, decree and establish it and command that what has thus been decided in the Council be promulgated for the glory of God.

@toddevoss Yes, when I worked for a parish as organist/assistant director this is what we did. Every Sunday and feast day Mass.

Otherwise, my opinion is that if a congregational Marian hymn is to be sung, the Offertory is the best time (and Communion the worst...which only reflects my opinion/preference for silent prayer at such a sublime moment). Unless, of course, it is a Marian feast day, for which it is especially fitting for a processional and/or recessional hymn to be sung.

We do this too, especially appropriate as the church is dedicated to 'St Mary of the Assumption'.