The decision has been taken ahead of Mother’s Day, which the Trust predicts will attract more people to its parks despite Government advice on social distancing.
Houses, cafes and shops belonging to the charity were closed earlier this week.
But following Boris Johnson’s announcement on Friday that all pubs, clubs and restaurants would be closed, the Trust has ramped up precautions.
The trust's director general Hilary McGrady said: “Despite our desire to keep our outdoor spaces open, the health and wellbeing of our staff, volunteers and visitors has to be our top priority.
“Having observed the numbers visiting our properties today I am no longer convinced we can maintain social distancing over Mother’s Day when numbers are likely to grow, and beyond.”
Ms McGrady added that the Trust would be taking measures to ensure that people did not lose their connection with nature, and that sites of natural beauty remained open “virtually”.
“We know that people are likely to need space and fresh air in the coming weeks and months and we will do all we can to provide access wherever possible,” she said.
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She added: “Over the coming weeks our digital platforms – our website, social media feeds, podcasts and video – will become even more important, ensuring the places of nature, beauty and history that we care for on behalf of the nation can remain open for business virtually while we are temporarily closed.
“We will also be ramping up our efforts to help people connect with nature wherever they are and to find moments of joy in the world around them. “