The Scottish government says people with second homes and campervans have been travelling to the Highlands in search of isolation in recent days.
They issued a travel warning, describing the actions of travellers as "irresponsible behaviour", reiterating the warning of Prime Minister Boris Johnson that people should instead stay home.
"The area has been overwhelmed with visitors. More worrying still is the significant crowding on the mountain summits and trails making it impossible to maintain effective social distancing."
The park is now calling for Boris Johnson and First Minister of Wales Mark Drakeford to provide "stronger measures" on unnecessary travel and social distancing so visitors do not flock to the park again.
A spokesman added: "Specific guidance is needed on what 'necessary travel' actually entails. We also call on all visitor and holiday owners to heed government advice and avoid all but essential travel, and to stay at home to stay safe.
"If no further steps are taken we will need to take drastic measures to protect the communities and health services in North Wales, such as shutting down car parks and trails."
Mr Drakeford has since urged people to stop socialising to try and combat the spread of the virus.
He said in a statement: "Self-isolation for those who have symptoms and social distancing by everyone is absolutely essential at this stage if we are to delay the spread of this virus and save lives.
"We all need to follow this advice now to protect each other and our families and to help ensure our NHS is not overwhelmed.
"Social distancing includes avoiding all but essential travel and if people don't follow this advice we will have no choice but to use powers to enforce it."
Rural economy and tourism secretary for the SNP, Fergus Ewing, said: "I am furious at the reckless and irresponsible behaviour of some people travelling to the Highland and Islands. This has to stop now.
"Let me be crystal clear, people should not be travelling to rural and island communities, full stop. They are endangering lives. Do not travel.
"Panic buying will have a devastating impact on the livelihoods of rural shops and potentially puts unwanted pressure on NHS services in our rural communities."
Further south, visitors and tourists have been told to stay away from beaches and other holiday areas to stop the spread of coronavirus and to avoid overwhelming health provisions in small towns.
Cumbria Police said the Lake District was among the places experiencing an "influx" of visitors and that people who do not live in the area should stay away.
"A national emergency shut-down of businesses and schools is not an excuse for a holiday," they added in a statement.
West Wittering Beach in Chichester was closed after the number of visitors on Saturday "far exceeded" expectations.
The estate owners said they had seen "an unacceptable movement of people" into the local community.
The MP for Cornwall, Steve Double, has also urged people not to go on holiday to the region amid the COVID-19 pandemic.
He said while he understands people's reasons for wanting to get out of cities, he said it was better to stay at home.
Mr Double told Sky News: "There are reports that roads into Cornwall are full of people with cars full of suitcases and top boxes. People are clearly coming down here.
"Regrettably and sadly some tourist parks are still taking bookings and still urging people to come here."
He added: "I think most reasonable people would consider a holiday at this time, when our country is facing such a huge challenge, as definitely not essential."
Visit Cornwall tourist board said on Friday that anyone who does not live in the area should postpone their visit.
They added: "We urge anyone considering travelling to Cornwall in the coming days and weeks to think carefully about the implications of being removed from your normal social network during the coronavirus period, the risk of not being able to travel back home as planned, to be aware that supermarkets are experiencing the same pressures here as elsewhere, and consider the added pressure on public services in a rural area which this travel may cause."
The warnings come after Mr Johnson ordered the closure of pubs, restaurants, theatres and other leisure places across the UK.
Schools across the UK have also now closed, except for children of key workers.
The moves are an effort to stop the spread of the illness, which has infected more than 5,000 people in the UK and more than 300,000 worldwide.
Almost 13,000 have died, including more than 230 in the UK.