The EFL, which comprises the three divisions below the Premier League, has seen 19 of its 36 games called off due this weekend.
Only four of the scheduled 10 Premier League games are going ahead with some postponements announced just hours before kick-off, causing further inconvenience for travelling fans.
Brentford manager Thomas Frank led calls for a circuit breaker earlier this week to allow the infection rate to stabilise and extra precautions at clubs to take effect.
However, Parry said there was no evidence to suggest the situation will improve quickly as Britain has posted record numbers of positive cases in each of the last three days.
“If we think a circuit breaker will help we will do it but, at the moment, there’s no scientific evidence to suggest that it will help,” Parry told the BBC.
“We’ve got to stick together, stay calm and keep making balanced decisions on the basis of the latest information.”
Full crowds are still permitted at matches that do go ahead in England, but supporters have to show proof of vaccination or a negative test at stadiums with over a 10,000 capacity.
The escalating numbers of cases have seen calls for more restrictions to be put in place by the British government.
London mayor Sadiq Khan on Saturday declared a “major incident” in the capital due to the tide of infections.
Parry admitted that the decision to stop the season may soon be taken out of the football authorities hands.
Football in England was stopped for three months between March and June 2020 when the pandemic first hit.
“There’s talk of the government having a two-week lockdown after Christmas so maybe that will be taken out of our hands,” added Parry.
“We are getting more information listening to the scientists, they need more data to try to work out exactly where we are going with this new strain. It is too early to tell.
“Next week we might be in a different position.”