Let’s say a parish has a patron saint, the diocese another and the religious order of the Priest in the parish has another patron saint. If these three saints fall on the same day which saint’s feastday is to be celebrated?
We celebrate the feast designated in the liturgical calendar of the Universal Church, not that of an individual parish, religious order, or private patron saint of a priest. He will always commemorate the Saint of the Day in the Roman Missal.
Would that be a day of fasting during Lent?
No, since the only fast days of obligation are Ash Wednesday and Good Friday. Maybe you meant to say “abstinence” and if so, then all Fridays in lent are days of abstinence, except for “solemnities” such as St. Joseph if it falls on a Friday; in that case, abstinence is dispensed.
Which feastday should the parishioner celebrate when saying the Divine Office at home?
Again, the saint of the day in the Roman Missal. The difference here, is that the person praying does not have to pray the
fullness of that saint’s feast day, as in the “Commons” unless it is a Church or Order specifically dedicated to that saint.
For instance, I am a Carmelite, and we have a more detailed Office for our patron saints of the Order, yet the Divine Office treats these days as ordinary memorials without any further fanfare (to use a mundane expression).