Hi Matthew, thanks for asking this question! I see that Joseph is ancestor of us both. I have never made much progress in properly identifying his parents.
Joseph's marriage record does not specify whether his parents were married. I have wondered whether he could have been born out of wedlock.
There are many potential Charles Gauthier entries in the PRDH that could be his father, but none of them seem to have lived in Trois Rivieres.
Josephe Berthiuame is a much less common name. There are two Josephe's about the right age, but again neither was born or lived in Trois Riveres as far as I can tell: Bertheaume-1 and Berthiaume-176.
The witnesses in Joseph's marriage record for the groom's side are three friends of his. The record states that his wife Catherine's parents were "of the parish of St Martin de Boulene, France", which is where they were born and married rather than where they resided at the time (her parents were present at the marriage ceremony). So I would think that Joseph's parents weren't necessarily living in Trois Rivieres at the time of his marriage in 1770.
Godparents for his two children were Jean Vienneau (Catherine's brother, Vienneau-43) and Francoise Masrol(?) (maybe Mazerolle-8), both baptisms recorded at the same time.
Edited to add: there is an Acadian Gauthier family, the family of Joseph Nicholas Gauthier (Gauthier-374). But no son Charles is known.