Pedestrians said traffic on roads through the park was making it difficult for them to keep the necessary 2m apart on narrow pavements.
The call follows anger that park owner Historic Environment Scotland has kept several car parks open despite Edinburgh City Council closing all of its car parks beside parks to discourage people from driving there.
Solicitor advocate Ross Mitchell tweeted: “Holyrood Park understandably full of people getting outside.
“Pavements are not wide enough to maintain safe 2m distance currently being recommended.
“Roads in park need to be closed in order for people to get outdoor exercise in the city.”
@thebonnieloon tweeted: “Holyrood Park must be the last park and car park open to vehicles.
“Why do they continue to ignore government/medical advice and facilitate the spread of #Covid_19?”
Also on Twitter, “Just Another Cyclist” said: “Someone must be able to do something about this? It's shocking.
“@HistEnvScot are actively stopping people exercise with social distancing.
“Holyrood Park car park and roads through the park remain open.”
@RangiferBirota tweeted: “Lots of people saying that it’s very busy with people going for a walk but the traffic confines them to narrow pavements and no social distancing possible.”
Living Streets Scotland, which campaigns for walkers, said they were being forced onto roads to keep the required 2m apart.
Director Stuart Hay said: “Many of us don’t have gardens so we need public parks to stay open so people can carry out their daily exercise in more spacious settings.
“However, we should not be making it easier for people to drive to them.
“Instead, we need to be making other areas more walking-friendly too.“We need fewer cars and lower speed limits to accommodate the fact that overcrowded footways and the need to keep 2m apart is forcing people into the carriageway.
“Cities worldwide are starting to reallocate road space temporarily to allow people to move more freely.
“We need to be following their lead to ensure people can carry out their daily exercise at a safe distance from others and free from road danger, rather than making further allowances for individual car use.”
A spokesperson for Historic Environment Scotland said: “All roads within Holyrood Park remain open with the exception of the High Road, which is closed for the annual toad migration.
“The car parking areas within the park - Meadowbank, Broad Pavement and Duddingston and Meadowbank triangle by St Margaret’s Loch - also remain open.
“Due to the layout and physical infrastructure of the park, it is not practical to close these car parks.
“We also wish to retain 24-hourr access for emergency vehicles should they be required to attend to an incident in the park.
“We recognise the park is an important space for local communities to undertake daily exercise in line with government guidance.
“However, we are urging all visitors to follow government advice by staying local, to avoid travelling to use the outdoor space and adhering to social distancing guidelines.
“As set out in the recently-published statement from Scottish ministers on access rights during Covid-19, we all have a duty to use public places responsibly and to comply with government guidance.
“We will continue to work closely with Police Scotland on this and keep the situation under review.”
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