Estates run by the trust have been besieged by huge groups of visitors over Mother's Day weekend, despite Prime Minister Boris Johnson’s pleas on social distancing.
A spokesman for the trust which, manages Clumber Park, Hardwick Hall and Wentworth Castle Gardens, said: “We urgently request people stay local, observe social distancing and do not travel.”
The trust previously said it would only close houses, cafes and shops, but has now closed car parks, parks and gardens to try and “restrict the spread of coronavirus".
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Hilary McGrady, Director-General at the trust 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 I am no longer convinced we can maintain social distancing.
“We have now sadly taken the decision to close our parks and gardens, in addition to our houses, shops and cafes, to avoid crowding that puts social distancing at risk.
“We know 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.
“We will also be ramping up our efforts to help people connect with nature wherever they are and to find moments of joy. We will be providing rich content and staying in touch with our members and followers throughout this time.”