your statement of what is happening in your parish is unclear, since it makes no distinction between adult Catholics who are being confirmed, baptized non-Christians who are making a profession of faith, and being confirmed and receiving first communion, and catechumens, the unbaptized, who will receiving baptism, confirmation and first communion at the same time.
Only the last category participate in the Rite of Acceptance in the Order of Catechumens (which happens before Lent, whenever appropriate, the Rite of Election (which belongs to the Bishop, or his delegate) which happens at the beginning of Lent, and the scrutinies, minor exorcisms and presentations, which happen during Lent. The ordinary time the elect (unbaptized) are initiated into the Church is at the Easter Vigil, unless there is an overriding pastoral reason that mandates another time (that change must be approved by the bishop).
when the first two classes of persons are confirmed depends on whether or not the bishop reserves confirmation to himself, which means it will be done at a time and place he decides, usually not Easter. Here it is Pentecost Sunday at the cathedral (or basilica church which is larger), or at the parish when the bishop makes his regular visit to confirm high school youth.
For the second group, baptized non-Catholics, the RCIA does provide optional parallel rites for them, such as the Rite of Welcoming, which may or may not be celebrated at the same time as the catechumens have their rites, and there is even a provision for a combined rite. They do not participate in the scrutinies, exorcisms or rite of election, by virtue of their baptism. They are brought into the Church through RCIA, the sacraments of initiation are RCIA, Rites of Christian Initiation for Adults. There education and preparation may follow a different schedule, however.
we just got a new forum for RCIA discussions, Evangelization.
If you give us a little clearer idea on what your bishop and priest actually said about each situation we can answer better.