It suggests that instilling a love of learning, and child well-being are more important to mothers and fathers than how well youngsters perform in government assessments.
The report, based on a survey of more than 2,000 parents, also finds that many do not choose primary schools for their children using test results.
Instead, they are more likely to rely on their own gut feeling when visiting a school, and recommendations from other parents.
The More Than A Score campaign group, which commissioned the poll, said the findings show the education system should focus on developing skills and cultivating a love of learning in children rather than “cramming them with facts”.
Under the current system, pupils at state primaries in England undergo a number of tests and checks, with a focus on maths and English.
Ministers have argued that these assessments help to ensure that children are grasping the basics and making good progress in the three Rs.
But the new survey suggests that parents believe that reports from teachers (48%) and seeing their child’s school work for themselves (35%) give more accurate information about a youngster’s progress than results of tests set by the Government (9%).
Nearly two-thirds (63%) of those questioned said pupils’ happiness should be used to measure how well a primary school is performing, while the same proportion (63%) thought that children’s progress in a range of subjects should be taken into account.
Some 61% said teaching that inspires a love of learning should be used to measure school performance.
Just 12% thought that tests taken by children under exam conditions should be used to assess how well a school is doing, with 29% saying testing not carried out under exam conditions should be taken into consideration.
The survey also asked parents what factors they took into consideration when choosing a primary school for their child.
More than seven in 10 (72%) said that how they felt when visiting a school would influence their decision, while 77% said teachers who care about their pupils and inspire them to learn would sway them.
More than half (53%) said they were influenced by Ofsted ratings, and 62% took account of good recommendations from other parents.
One in four (25%) said they were influenced by good results in tests set by the Government.
Sara Tomlinson, of More Than A Score, which is campaigning for changes to primary school testing, said: “Parents want to be reassured about the quality of their children’s education. They want a broad curriculum and inspiring teaching. They do not want their children to be subjected to unnecessary testing purely for the purposes of gathering data to create league tables.
“We need to ensure that our education system is one that focuses on developing skills in our young people and cultivates a love of learning for life, not simply on cramming them with facts.
“The Government must now listen to those who know children best – educators, experts and, above all, parents.”
Currently, pupils in England sit national curriculum tests (known as SATs) in maths and English in the final year of primary school. These results are published and often used in holding primaries to account.
In addition, children take a phonics check in Year 1, and a times table check in Year 4.
A baseline assessment is due to be introduced in all of England’s state primaries from this September.
It will be taken by pupils aged four and five, in Reception, with the results used to measure the progress they make throughout primary school.
There are also currently national curriculum tests in English and maths for seven-year-olds.
In 2017, the Government announced proposals for these to be made non-compulsory from 2023, following the introduction of the baseline check.
Primary school SATs have long been controversial, with critics arguing that they put too much pressure on young children, while supporters say they help identify pupils who need assistance and ensure schools are helping youngsters grasp the basics.
A Department for Education spokesman said: “Key stage 2 tests have been central to raising standards, helping to ensure children leave primary school with a secure grasp of the fundamentals of reading, writing and mathematics.
“As part of a broad and balanced curriculum this helps lay the foundations for success at secondary school and beyond.
“We trust teachers to administer these tests in an appropriate way and so they should not be a source of stress for children. The tests enable teachers to track pupils’ progress, helping to make sure they stay on track to fulfil their potential throughout school.”
Paul Whiteman, general secretary of the National Association of Head Teachers, said:
“Some schools find it impossible to balance the competing demands of government policy and inspection.
“Pupils and staff alike would be better served if policy, accountability measures, inspection and improvement support are properly aligned.”
– The online YouGov poll questioned 2,028 parents of children aged three to 13 not at independent schools or pupil-referral units in England between November 12 and 18.