Around 20 children with special education needs (SEN) are currently without a school place for September.
The Education Authority (EA) has said there is no guarantee the children will receive one before the school term begins.
Cynthia Currie from the EA said there were not enough appropriate school places for children with SEN.
It comes as some families held a protest at the EA's headquarters in Belfast on Wednesday to call for answers.
The EA met with the parents to discuss their issues and potential next steps.
The BBC reported that when asked if the EA could guarantee those children a place, Ms Currie replied: "We certainly hope that would be the case, but it would be disingenuous for me to give you an absolute categorical answer without the work that needs to happen between all the parties".
Among those still waiting for a place is Iarla Murphy (3), who has Down Syndrome.
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His mum Kathy told The Irish News that her son "was due, like many of his peers, to start nursery school in September".
"However, unlike most of his peers, Iarla has not been allocated a nursery place," she said.
Ms Murphy said he should have received a statement of SEN, which sets out a child's needs and the support they should have in school, from the EA 18 weeks ago.
But she said the process has not yet finished.
"Simply stating that they are working hard to place all nursery children, is not, in our opinion, an adequate response to any specific inquiry," she said.
"In short, it has been impossible to communicate with anyone from the EA or to get any genuine answers regarding our own child."
An EA spokesperson said they welcomed the opportunity to meet with families to "listen to their concerns in relation to SEN placements".
“The EA continues to work to ensure all children with a statement of SEN which fully meets their needs to ensure that they are happy, learning and succeeding, and this remains our top priority," he said.
“As of 15 August - 4185 children of 4205 in transition years (new nursery, nursery to P1 and P7 to Year 8) –approximately 99.5 per cent - have a confirmed placement for September.
“As of 15 August 2023, there are 20 children currently awaiting a place for September and a significant programme of work continues to ensure that all children with a statement of SEN have a place which meets their needs.
"We fully expect to confirm more placements in the coming days thereby significantly reducing the number of children and families still waiting on confirmation of a place.
“We fully understand the anxiety of parents/carers and the impact on those children and young people awaiting confirmation of where they will attend in September and are firmly committed to continuing to keep families fully informed during August.”