Opposition parties on Friday felt it was Basic Education Minister Angie Motshekga's responsibility to ensure textbooks were delivered timeously to Limpopo schools.
The Congress of the People said the department continued to be in turmoil and fail pupils and teachers.
Following Motshekga's announcement on Friday afternoon that she could not guarantee that another textbook crisis would not happen again on her watch, the Democratic Alliance said it felt it was her responsibility to ensure it did not.
"If Minister Motshekga cannot guarantee this, then she must resign," DA MP Annette Lovemore said in a statement.
Earlier on Friday, Motshekga denied blame for a delay in the delivery of textbooks, and said she would not quit.
"I have not considered resigning," she told reporters in Polokwane.
"The best I can do is sort out what I started. I won't jump ship."
A number of factors contributed to the delay, including cash flow and administrative problems, she said at a televised press briefing, maintaining her innocence.
"I trust my officials. I had no reason to doubt them. I won't put my head on the block... and leave my children motherless. The head of the institution, he has the ultimate responsibility. The buck stops with him."
She said it was apparent the previous administrator underestimated the magnitude of the deliveries. He was instructed in February to order textbooks, but delayed doing so.
"This put us under extreme pressure... In April, there still was no books, so I replaced him [sic]," the minister said.
In May, the High Court in Pretoria ruled the department's failure to provide textbooks violated the Constitution. The application was brought by rights organisation Section 27.
Judge Jody Kollapen ordered the department to devise a catch-up plan to remedy the consequences of the delay, and to supply the affected schools with textbooks by 15 June.
The department failed to meet the court's deadline, but Motshekga said her department had met with Section 27 and agreed to move the deadline to 27 June.
The minister said the department had accepted an offer from a company called UTI to help them deliver the textbooks. The company provided them with resources including trolleys, horses and trailers. A recovery plan had also been put in place so children could recover lost time.
"It is a comprehensive catch-up plan," she said.
This included a winter programme for Grade 12 pupils and Saturday classes. Other grades would have their own plan, depending on what textbooks they did not get.
The minister maintained that not all learning was lost as some topics in the new curriculum were also in the old curriculum.
The national department took over the running of Limpopo's education department in December following maladministration.
The Democratic Alliance on Thursday offered a fleet of vehicles to the minister to help deliver the textbooks, but the offer was declined.