There is no specific prohibition against holding hands during the Our Father, or any other time at Mass for that matter, either for the Novus Ordo or the TLM.
However, there is also no provision to ask or invite people to do so. Were a priest or anyone else to do so during Mass he/she/? would commit a grave liturgical abuse.
Congregations of total or near total strangers might be spontaneously driven sincerely to hold hands in some circumstances.
But – and perhaps it is a lack of something on my part – I cannot see this hand holding stretch exercise across aisles, for example, as a regular practice as anything other than contrived sentimentalism which distracts us from the transcendent nature of God Almighty and the meaning of the petitions in the Our Father.
On a related note, during the Our Father the faithful are not to use the so-called “orans position” (“praying position” with hands extended, open), which is the proper hand position of the priest celebrating the Mass. Even worse is when they hold that position after the Our Father through the (Protestant) addition that follows. The orans position is reserved for a certain liturgical role (read: priest – not even deacons). That position of extended hands is not appropriate for the lay faithful in the pews.
We must not mix or confuse liturgical roles. Lay people have their own dignity without trying to jazz them up by – and how condescending is this? how clericalist? – allowing them to do what the priest does. That’s the worst sort of clericalism and it is always used by libs, isn’t it? The subtle message given, when roles are purposely confused for the sake of “active participation” or “getting the laity involved”, is really “You aren’t good enough on your own, so I’ll let you do something that I can do.” Grrrrr. But I digress.
So, I repeat: I am unaware of a prohibition of holding hands during Mass. Spontaneous hand holding? Fine. It must never be invited or imposed by someone with a microphone anywhere near the altar or by anyone in the pews.