Achieve, Inc. released a survey that contains some interesting data. Since they are the group behind Common Core and are Gates-funded, I think the majority support indicated in their poll as utter hogwash however.
They explain the methodology that makes the Common Core support polling suspect:
On behalf of Achieve, Public Opinion Strategies with Greenberg Quinlan Rosner Research conducted a national survey of 1,200 public school parents, with at least one child in grades 3–12 with oversamples of 300 each of Hispanic moms; African-American moms; and suburban moms with a household income of +$80,000.
The survey was conducted in May 2016 and has a margin of error of +2.83%. The margin of error for the oversamples is +5.7%.
Oversampling Hispanic moms and African-American moms skew the results as minority groups in urban areas have tended to poll higher in their support of Common Core.
What this does do is show that it appears more parents will opt their students out of assessments in the 2016-2017 school year as you can see below. The numbers double with Hispanic and African-American moms.