Boris Johnson has warned “tougher measures” could be introduced if people do not take the government’s coronavirus advice seriously.
The PM thanked people for making sacrifices but said people must follow social distancing guidance.
“If you don’t do it responsibly… we will have to bring forward further measures,” he said.
It comes as the number of UK deaths reached 281, including a person aged 18 with an underlying health condition.
They are thought to be the youngest person with the virus to have died in the UK so far.
The rise of 48 deaths since Saturday includes 37 in England, seven in Wales, three in Scotland and another in Northern Ireland.
The NHS said all those who died in England in the past day were in vulnerable groups including with underlying health issues.
It comes as the NHS in England has identified 1.5 million of the most at-risk people who should now stay at home for 12 weeks.
The PM told those people to “shield” themselves, adding it “will do more than any other single measure that we are setting out to save life”.
Speaking at Downing Street’s daily news conference, Mr Johnson told people going to parks they “have to do that responsibly”.
It comes after pictures showed people across parts of the UK visiting parks and open spaces in large numbers over the weekend.
Snowdonia National Park said the area “experienced its busiest ever visitor day in living memory” on Saturday, with other beaches and mountain summits busy.
“Don’t think fresh air in itself automatically provides some immunity,” Mr Johnson said, adding that even if people think they are invulnerable, “there are plenty of people you could infect”.
“Take this advice seriously, follow it, because it’s absolutely crucial.”
“My message is you’ve got to do this in line with the advice, you’ve got to follow the social distancing rule – keep 2m apart.”
Asked whether stricter measures could be introduced, Mr Johnson added: “I don’t think you need to use your imagination very much to see where we might have to go, and we will think about this very, very actively in the next 24 hours.
“It’s so important that that pleasure and that ability is preserved but it can only really be preserved if everybody acts responsibly and conforms with those principles of staying apart from one another and social distancing.
“If we can’t do that then, yup, I’m afraid we’re going to have to bring forward tougher measures.”
Some parks have already announced they will be closing. Essex County Council will close all its country parks from 20:00 GMT, while earlier Richmond Park in London closed to traffic on Sunday, although those on foot and cyclists were still allowed.
Earlier, Scottish First Minister Nicola Sturgeon said the new measures to prevent the spread of coronavirus should not be considered “optional”.
“Life should not feel normal,” she said, and if it did you should ask “if you are doing the right things”.