I used to teach middle school math in the Boston area, and several of my seventh grade students did the Kumon program outside of school. Kumon math appealed to students who were well-organized and who liked neat, tidy rows of computational problems with one right answer per problem. Unless the Kumon program has changed in the last few years, it involves worksheets of computation problems that get progressively more difficult. When you get a high percentage of problems on a worksheet correct in a certain time period you can progress to the next worksheet. Working fast is definitely stressed. I do not know if any particular strategies for solving the problems are advocated or taught, but I'm definitely curious, if anyone can answer that question.
TERC definitely advocates the development of efficient strategies for computation-- I think the question is WHEN. I would try to explain to parents that it's important for students to have a certain amount of time and practice in their development towards efficiency. Kumon's emphasis on speed could cause them to jump to Mom's method without real understanding. But maybe Kumon Math wouldn't hurt if they need practice on what should have been learned long ago. For instance if a third grader takes too long to come up with addition facts or a fifth grader with multiplication facts the practice and the need for speed on Kumon worksheets might push them to master quick recall.
I would be extra concerned if students are doing Kumon-style worksheets on the topic of computation with fractions and decimals. My experience teaching math at the middle school and high school level is that when students don't have a good conceptual understanding of rational numbers (fractions, decimals, percents) they cannot succeed in algebra and beyond. It's so important to do the conceptual work around fractions and the Investigations fraction units do this brilliantly. Parents might be interested to know that in France they don't teach adding fractions with different denominators until 8th grade! They don't think students are ready developmentally for the difficult concept of finding a common denominator until then. (French schools teach a lot of simple algebraic ideas at younger grades than the U.S., but that's another story.) In other words, what's developmentally appropriate to teach at a certain grade is certainly not commonly agreed on. But parents often think math skills should be taught on the same timeline as when they were kids. So Kumon Math is attractive to them if their kids' teachers don't agree with their timeline.
Try to get the parents to give students a year later than they are used to to master computational skills. They should understand that doing more work with the concepts first takes time (but this is problematic if you're facing middle school teachers or standardized tests that don't agree.)