Does Rev. Parris care about his 'name'?
Reverend Parris' name is all he seems to care about. When Betty is "ill" at the beginning of the play, Parris is more worried about what people are saying than about what's wrong with Betty. He drills Abigail about what the girls were doing in the forest and her own reputation around the village because it reflects on him. Parris is paranoid that people are always talking about him or are out to force him out of the church. He is quick to believe the girls' accusations of witchcraft because it relieves him of the burden of having to justify what he saw the girls doing in the forest. Throughout the witch trials, Parris is only worried about making himself look good and look important. He cares nothing about what is happening to the people of his church and his village. By the end of the play, Parris is a broken man.
The Reverend Samuel Parris is very concerned about his "name." By that, we mean that he is very worried about his reputation -- the way people think of him.
This is partly because he is the town's minister. In Puritan times, this was a very important position and he would have felt the need to have everyone trust him. It is also partly because of his own ego. He really wants everyone to look up to him. This is why he demands things like golden candlesticks for the altar. This is also why he is so worried about Betty and her apparent connection to witchcraft.