Burton Borough School, in Newport, may remain closed for the rest of the week and pupils were being taught online on Monday as a result of staff having to isolate.
The school has written to all parents about the situation after 12 cases were recorded up to Friday.
A parent with children at the school, who did not want to be identified, said she disagreed with the Government's plans to lift Covid-19 restrictions from the end of next week.
"The school is closed due to a spike in cases at a time when the Prime Minister is thinking about relaxing the rules. School children don't need to wear face coverings anymore.
"Since that stopped cases have been rising. I'm concerned about what will happen when there is further relaxing of the rules if small towns like Newport which is among the lowest affected is seeing a rise in cases among children.
"I think relaxation of the rules any further will be catastrophic.
"The Government shouldn't be bowing to pressure from certain groups to roll back restrictions.
"We hear about children's education being affected, but the emphasis is being put on pupils wellbeing. We are not hearing much about the fact that right now people are fighting for their lives.
"This pandemic is not going away unless strong measures are in place."
In the letter also posted on the school's website head Krissi Carter stated: "It is with regret that I have to inform you that Burton Borough School will be closed on July 5, 2021 due to operational reasons following the spike in Covid-19 cases which has caused a number of staff to isolate.
"We are still working with the health protection hub and local authority education services on whether school will be open after Monday for the rest of the week.
"Since I last wrote to you, we have had some additional cases, bringing the total number to 12. We have worked incredibly hard all day today and yesterday evening to get the necessary information to all families affected and I thank everyone for being so understanding and patient. We will continue to work throughout the day tomorrow and inform you of any updates where appropriate.
"All lessons will be on MS Teams on Monday.
"Once again, thank you for your continued support – it is much appreciated.
"Take care and stay safe."