I'll check for you, but we moved around a lot while raising our children. I always had to turn in the name of the parish where the kids were baptized when they received other Sacraments, so that information could be sent back to the Baptizing Parish and the information could be added to the Baptismal Certificate.
When I retrieved a fresh copy of their certificates when the kids got married (a requirement for a Catholic marriage), the information was all there. I am told that this is done so that the Church has a record at the time of the individual's marriage or Holy Orders.
So, the answer may be that the "confirming" parish does not keep the record; it is sent to the "baptizing" parish, but I'll check with my Parish office and ask the question.