As a halachic convert to Judaism, I know that I am not supposed to enter a church or other non-Jewish place of worship. However, my parents are religious Christians, and I wonder if I am allowed to enter a church for the funeral of one of them, as this would be respectful to my parents. Does the Mitzvah of honoring one’s parents outweigh the prohibition in such a case?

You are not permitted to enter a church for this reason. Honor their memory in other ways- by visiting the home, giving a donation in their honor with a notice of the donation sent to the family etc.