> Is there evidence the amish abuse their children at a rate higher than the rest of society…
Where to start? The Amish stance on corporal punishment of children is well-known. "Spare the rod, spoil the child" (as Proverbs 13:24 is commonly misquoted) applies. Hitting children is common in Amish households and schools. Sources on this range from atheist neighbors[1], to friends of the Amish[2], to books that praise the Amish[3]. And speaking of Amish schools: They end at the 8th grade. People like to think that Amish teenagers have a choice about leaving the community, but what choice do they have when their prospects in modern life are so limited by their lack of education?
To frame it more concretely: If you had a neighbor who denied their children cold drinks or air conditioning on hot days, who hit them for questioning religion, and who took them out of high school so they could do work, is there any doubt that you would call child protective services? It flabbergasts me that the Amish get a pass here.
> …and/or other cultural enclaves such as tight-nit immigrant groups?
This is moving the goalposts. It doesn't matter whether other groups also abuse their children. "But FLDS groups abuse their kids too!" is not an excuse that our society tolerates. In fact, with the FLDS, we sent in SWAT teams and arrested the church leaders. Sadly, the Amish seem to be grandfathered in.
The child abuse is the issue that gets the biggest rise out of me, but it's just the tip of the iceberg. There are many other ways in which the Amish needlessly inflict suffering on each other. Due to their piety, Amish positions on topics such as birth control, homosexuality, and the rights of women shouldn't surprise you.
1. http://freethoughtblogs.com/butterfliesandwheels/2012/09/wha...
2. http://amishamerica.com/how-do-amish-discipline-children/
3. Amish Roots: A Treasury of History, Wisdom, and Lore p. 111, section titled "Disobedient Children"