I have always been told that this is the one exception to the rule that Catholics bring up their children as Catholic. I have no authoritative documentation to back up this claim. I was taught that it is because the rules of the Orthodox Church are stricter.
There is nothing in the CCC or Vatican II documents to support this position.
In all other cases, the non-Catholic parent, regardless of faith or religion, must agree to bring up the children Catholic. Form may be waived if the Catholic marries a minister of a mainline Protestant church. I did not learn of this possible dispensation until after my daughter married a Presbyterian minister. Then again, she had already left the Church before meeting him.
Marriage is a Sacrament that the man and woman bestow on one another. The priest, or deacon, is the prime witness. The priest is necessary if the wedding is within the celebration of a Mass, the only way that I was willing to marry.