The Educational Institute of Scotland (EIS) said it supports the right of parents to opt out of the tests – officially called Scottish National Standardised Assessments.
Pupils starting school are assessed on literacy and numeracy as part of an SNP effort to improve academic standards.
But education campaigners have warned of the negative effect that the exams are having on some youngsters, saying some pupils have been reduced to tears.
Sue Palmer, a former teacher, is leading the Upstart campaign, which opposes the testing. She believes young children should learn through play: “Our message to parents, from a point of view of children’s educational success, is the best thing is play, and testing is counter-productive.
“In the case of children who are less able to read or write at four or five, and who are pressured into doing it, it could be damaging.
“If you feel this testing will be harmful for your child, don’t do it. And even if you don’t but you feel it’s harmful for other children, then why do it? We hope they will withdraw them from the tests.”
The EIS said: “The EIS supports the right of parents to opt their children out of the P1 standardised assessments but clearly that is a parental decision.” A Scottish Government spokesman said: “There is no statutory right for parents to withdraw their child from any aspect of schooling other than some parts of religious and sexual education.
“However, as the Scottish Government has always maintained, if a parent does not wish their child to take part in an aspect of teaching and learning, they should discuss this with the school, who have the discretion to decide whether the child takes part or not.”
Council umbrella body Cosla said: “If there is a concern held by parents then headteachers and councils will respond to those concerns.”
Earlier this month, improved pass rates for Highers were reported, although National 3, 4 and 5 pass rates fell to a four-year low.