One mother told the BBC that her child, who loves reading, was unable to finish the paper.
A head teachers' union said even staff had struggled to understand the questions, and it would be raising concerns about the paper.
The Department for Education (DfE) said it worked to ensure that "all tests are appropriate".
Some parents said on social media that their children were "distraught" after the paper, which is part of a series of national curriculum tests known as Sats.
A head teacher in Cheshire wrote to her MP calling for Sats to be scrapped after her primary school pupils were left "broken".
Jill Russell, from Cumbria, said her daughter, Pashley, was "very close to tears" when she picked her up from school on Wednesday.
Pashley, who is autistic, loves reading and is the subject ambassador for English in her school. She had been worried about Sats, but reading was "the one she was least concerned about".
"She usually ends up having a lot of extra time left over, and she said 'I don't think I got to the end of the paper.... I didn't understand a lot of it. It didn't make sense'," Ms Russell said.
"It's definitely made her more anxious about going back in today [Thursday]."
Ms Russell thinks it is "good, in a way, to have some kind of tests" before GCSEs and thinks Pashley's school is "fantastic", but feels that schools in general are under too much pressure to perform well in Sats.
"It kind of feels like they're being taught how to pass the test, as opposed to being taught, and then the test is an addition," she said.
The government has advised that the content of the test paper should not be published until all Year 6 pupils have had the chance to take it.
Sarah Hannafin, head of policy for the National Association of Head Teachers (NAHT), said the union was "very concerned" about the paper.
"Members have told us that the choice of texts was not accessible for the wide range of experiences and backgrounds children have and the difficulty was beyond previous tests, leaving children upset and with even staff struggling to understand the questions," she said.
She said the NAHT would raise the concerns with Standards and Testing Agency, which delivers assessments, and Ofqual, England's exams regulator.
Dr Mary Bousted, joint general secretary of the National Education Union, said the Sats this week had been "a punishing experience for many pupils and staff".
She said that children who do not meet expected standards in results this year "will take this demotivating label with them into their secondary schools".
"This is not a system that is concerned about children and their learning. There are better ways of assessing pupils," she said.
A DfE spokesman said Key Stage 2 assessments "play a vital role in understanding pupils' progress and identifying those who may have fallen behind".
"Our test development process is extremely rigorous and includes reviews by a large number of education and inclusion experts and professionals, including teachers, and we trial tests with hundreds of pupils over several years to ensure that all tests are appropriate," he said.
"It's important that schools encourage pupils to do their best, but preparing for these exams should not be at the expense of their wellbeing."
Last year, 59% of Year 6 pupils met the expected levels in reading, writing and maths - down from 65% in 2019.
The national curriculum tests were cancelled in 2020 and 2021.