To understand Parris' motives in keeping "unnatural causes" as the explanation for Betty's illness secret, you have to understand a bit of his background. He was the minister in Salem at the time, but there were quite a few people who did not like him. Some people had wanted a different minister, and Parris' fiery sermons had alienated and disgusted many members of his congregation. So, if his daughter is sick because of "unnatural causes," that means that witchcraft would be found in his very house. Also, he had caught Betty and Abigail dancing in the woods, and was worried that the girls themselves were the perpetrators of the witchcraft. So, Parris later explains to Abigail that if people knew that witchcraft had impacted his household, that his "enemies will...ruin me with it." He also explains that
"my ministry's at stake...I have fought here three long years to bend these stiff-necked people to me, and now, just now when some good respect is rising for me in the parish, you compromise my very character."
If Susanna goes around telling everyone that Betty has been bewitched--Betty, who was a minister's daughter--it would cause an uproar in the town, and anyone who didn't like Parris and wanted to harm him could use that as proof that he wasn't a divinely inspired minister. It would ruin his reputation and undo any grounds he had made with the people in the town.