Infectious disease expert Dr Mike Tildesley said it is "possible" to keep the reproductive number - or R value - of the virus below 1 with schools open, but everyone must continue to follow "all the other rules".
Schools in England are due to reopen from Monday, a move Education Secretary Gavin Williamson said marks the "beginning of the road back to normality".
Dr Tildesley warned parents: "When you're dropping your children off you need to maintain social distancing.
"Just because you're not in the home with your young children don't use it as an excuse to go out and mix with other people that you otherwise wouldn't have done.
"It's possible with schools open we can keep the R number below 1 but if we are going to achieve that we all need to keep following all the other rules."
Dr Tildesley told Times Radio that falling Covid-19 rates are most likely to be due to lockdown measures, and the impact of vaccinations "hopefully is yet to come".
He said: "I think most of the reason the numbers are going in the right direction now is still due to lockdown.
"I think we haven't quite seen the impact of vaccinations, probably start to come in round about now and having a little bit of an effect, but most of the effect thus far actually is probably the fact we have been under severe restrictions since the start of January."
Deaths from coronavirus have fallen by 41% in a week, while hospital admissions have seen their fastest ever fall, the Health Secretary said on Friday.
Matt Hancock painted a positive picture regarding the state of coronavirus infections in England in a press conference, with the average number of cases - 6,685 per day - the lowest since late September.
Dr Tildesley, a member of the Government's SPI-M modelling advisory panel, said: "We do need to get this balancing act correct and we need to open up at the rate of vaccinations and keep the R number in check, as it were.
"Definitely things are moving in the right direction but the next few weeks are going to be crucial for us to monitor what happens when schools open.
"Hopefully we can keep everything down and most importantly we can prevent seeing a rise in hospitalisations."
Mr Hancock said on Friday there were still 12,136 people in UK hospitals with Covid-19, which was "too high", but the seven-day average for the number of new admissions to hospital was 900 - "the lowest since October".
More than a million people in the UK have received both doses of a Covid-19 vaccine, while almost 21.4 million people have had one dose.
Dr Tildesley added he would have preferred to see primary pupils to return to school first "and maybe a slight delay for secondary school children going back".
Those returning to secondary schools in England will find safety measures including face coverings in classrooms while rapid coronavirus tests have also been introduced.
Scottish schools began a phased reopening last month.
Meanwhile experts said vaccines need to be redesigned to prevent widespread transmission of coronavirus variants such as that first found in Manaus, Brazil.
A mystery carrier of the P1 variant of concern was found in Croydon, south London, after a team of 40 people, armed with just a single bar code and limited test data, spent days tracking them down, it was announced on Friday.
But Professor Ravindra Gupta, a member of the New and Emerging Respiratory Virus Threats Advisory Group (Nervtag), said: "There will be people out there who have not been tracked and traced who have the variant and who may have transmitted it."
Prof Gupta, an expert in clinical microbiology at the University of Cambridge, said it is "always a possibility" that the Manaus P1 variant could become dominant in the UK but it is "unlikely at the moment" due to low transmission rates.
He told Times Radio: "We need to have redesigned our vaccines for the coming year with some of the key mutations present in those vaccines so actually we can prevent transmission of those variants if they were to take off in the coming months."
He added it was "inevitable that the list (of mutations) will grow because in many areas of the world transmission is still significant and transmission equals chronic infections".
"On the other hand, we can take comfort in the fact that the virus is doing very similar things across all of these variants, so there are some very common themes coming along and very common mutations, so that helps us to design the next generation of vaccines," he added.
From Saturday, businesses of all sizes, including those with fewer than 50 employees, will be able to register to order lateral flow tests, which can produce results in less than 30 minutes, for their workers.
From Monday, people travelling internationally from England will need to complete and carry a declaration to travel document downloaded from the Government's website.
Those who fail to comply with the measure - announced 40 days ago - face being fined £200, with holidays banned under current lockdown rules.