Perhaps there are song parodies that might be objectionable material in general, but most of the tried and true carols that we know well can be sung by Catholics.
However, if the question you are asking is if these tunes can be sung at mass? Then the answer is no. Secular songs do not fit with the liturgy at any time and we come to mass to celebrate the coming and birth of Christ. So we only select songs that reflect the true meaning of Christmas for the mass. In fact, even traditional Christmas carols are not sung at mass during Advent. Songs like “O Come, O Come Emmanuel” are sung to help create the longing for Christmas in our hearts.