Buddhism has been brought to the west within the last 60 years, and though it has gained popularity, it isn't really part of our culture. Most American Buddhists converted as adults. There is extremely little written material in English on how to raise Buddhist children or parent in a Buddhist way. Christians have all kinds of books, games, homeschooling curriculums, parenting courses, and church and other social groups. But the same support and guidance simply does not exist for English speaking westerners.
I have not changed my parenting style since becoming a Buddhist. There are no Buddhist teachings which contradict AP parenting and gentle discipline philosophies, so I feel all aspects of my life are lining up well.
I really don't have enough experience with "Buddhist parenting" per se to feel comfortable sharing advice on any "how to"'s. I am just trying to figure it out as I go along!