Both Proctor and Parris see themselves as authority figures but for different reasons. Despite just being a farmer, Proctor sees himself as having authority over others, especially over his own family. His sense of authority actually even makes him act like a bully. We especially see his bullying characteristics in the way he treats both the servants Abigail and Mary Warren. Similarly, Parris also believes he has authority but more specifically authority over the church as the town's minister. He believes that he was called by God to lead the townspeople as a minister, which gives him authority. For that reason, he becomes angered and feels bullied when people like Proctor threaten to vote to have Parris dismissed simply because they don't like his sermons. We clearly see Parris's sense of authority in the passage:
I am your third preacher in seven years. I do not wish to be put out like the cat, whenever some majority feels the whim. You people seem not to comprehend that a minister is the Lord's man in the parish; a minister is not to be so lightly crossed and contradicted ... (I.i.)
But what's more, this passage also reveals how townspeople like Proctor are bullying Parris. Hence, ultimately, Proctor and Parris have contradictory thoughts concerning who is in authority and who is not. Proctor feels that he has authority, while Parris feels that authority has been granted to him.