Since you refer to parish registration, I suppose you’re living in the United States? The idea of parish registration doesn’t exist in any other country, but only in the United States. Canonically speaking, parish registration doesn’t actually mean anything. One’s parish is determined by where they live, and the only way to really change parishes is to move.
I realize I wasn’t clear enough earlier. Regarding WHERE Catholics should get married, canon 1115 states that marriages should be celebrated in a parish where either of the spouses has a domicile, quasi-domicile, or month-long residence, or in the parish where they actually reside. Marriages can only be celebrated elsewhere with the permission of the diocesan bishop or the parish priest.
Canon 102.1 clarifies that a Catholic has a domicile if he resides in a parish with the intention of staying there indefinitely, or if he has ALREADY lived there for five full years.
Canon 102.2 states that a person has a quasi-domicile if he has lived in a place for three months, or intends to stay for at least three months.
One month is the bare minimum.
If this situation met the above requirements, then a marriage would have been celebrated validly.