One of the organizers, Christine McGoey, has two children in the local school system, and both will be sitting out the tests.

“I’m refusing because we’re taking a stand against this deeply flawed policy,” Ms. McGoey said. Parents who object to the tests have been communicating their concerns to local officials, she said, “but they’re just not listening.”

“I feel like the only thing left to do is just say no,” she said.

While these parent groups are vocal and get a lot of attention on social media, it is difficult to know how many people actually stand with them or how many will refuse the tests for their children. In the Millburn School District, about a dozen students had refused as of last week; in Bloomfield, 97 of about 6,200 students had opted out.

“Our board of education has taken a very strong stance against standardized testing,” said Salvatore Goncalves, superintendent of the Bloomfield School District, who added that there was “no doubt” children were being tested too much.

In New York last year, the state’s second year of Common Core-aligned testing, 49,000 students did not take the English test, according to the State Department of Education, while 67,000 skipped the math portion — numbers that include not only refusals, but also any student who did not take the tests for a “known valid reason.” Statewide, 1.1 million students took the assessments.

Education officials and some experts say the new tests, which require more writing and critical thinking, as opposed to filling in bubbles on an answer sheet, are a vast improvement over previous exams. And the Partnership for Assessment of Readiness for College and Careers will give detailed, individual feedback to children in New Jersey for the first time. But in virtually every state, the tests will be tougher than the old ones, stoking fears that scores will plummet, as they did when New York began using its new exams.

There are generally few repercussions for students who do not take the tests, but if more than 5 percent of the student body at a given school or district opts out, that school may risk certain consequences, like greater monitoring or the loss of money for needy students.