And from EWTN:
Concerning holding hands in the Eucharistic Liturgy the Congregation for Divine Worship in Rome responded as follows:
QUERY: In some places there is a current practice whereby those taking part in the Mass replace the giving of the sign of peace at the deacon’s invitation by holding hands during the singing of the Lord’s Prayer. Is this acceptable? REPLY: The prolonged holding of hands is of itself a sign of communion rather than of peace. Further, it is a liturgical gesture introduced spontaneously but on personal initiative; it is not in the rubrics. Nor is there any clear explanation of why the sign of peace at the invitation: “Let us offer each other the sign of peace” should be supplanted in order to bring a different gesture with less meaning into another part of the Mass: the sign of peace is filled with meaning, graciousness, and Christian inspiration. Any substitution for it must be repudiated: Notitiae 11 (1975) 226. [ Notitiae is the journal of the Congregation in which its official interpretations of the rubrics are published.]
While this addresses the holding of hands at the Sign of Peace the reasons given apply also elsewhere in the Mass, including at the Our Father.
It is an inappropriate “sign,” since Communion is the sign of intimacy. Thus, a gesture of intimacy is introduced both before the sign of reconciliation (the Sign of Peace), but more importantly, before Holy Communion, the sacramental sign of communion/intimacy within the People of God.
It is introduced on personal initiative. The Holy See has authority over the liturgy according to Vatican II’s “Constitution on the Sacred Liturgy” #22 and canon 838 of the Code of Canon Law .
This gesture has come into widespread use, often leaving bishops and pastors at a loss as to how to reverse the situation. For individuals, I would recommend closed eyes and a prayerful posture as sufficient response, rather than belligerence. Most laity, and probably many priests, are blind to the liturgical significance of interrupting the flow of the Mass in this way. It is not necessary to lose one’s peace over this or be an irritation to others. Some proportion is required. If asked why you don’t participate, simply, plainly and charitably tell the questioner of your discovery. If some chance of changing the practice is possible talk to the pastor or work with other laity through the parish council. You can also write the bishop, as is your right in the case of any liturgical abuse not resolved at the parish level. If your judgment is that no change is possible then I believe you are excused from further fraternal correction.