The roads through the park are currently closed at weekends between 8.15am and 3pm and the High Road around the back of Arthur’s Seat has been closed altogether since March last year.
But the Car-Free Holyrood Park campaign wants the vehicle ban to be made full-time.
It says for over 20 years people have been pressing Historic Environment Scotland, which manages the park - and its predecessor Historic Scotland - over problems of speeding and dangerous driving, narrow pavements next to busy roads, incomplete cycle paths, only one pedestrian crossing for the entire park and the long-standing ban on commercial vehicles being regularly ignored.
And it argues the current lockdown provides an excellent opportunity for Historic Environment Scotland to trial more road closures.
A campaign spokesperson said: "With Scottish Government and the council both recognising the need to prioritise walking, wheeling and cycling for our health and our environment, we think that now is the time to expand the successful weekend closures of the roads in Holyrood Park to full-time and end motorised through-traffic in the park.
"There are immense benefits to closing the park roads to motor traffic: it opens that space for people to walk, wheel and cycle making it a much more relaxed visit without having to worry about traffic, with reduced air and noise pollution, safe spaces for children, more space for those pushing buggies or using mobility aids, safe homes for wildlife, an end to speeding and dangerous driving and increased capacity and physical distancing at an extremely popular local destination.”
Lothian Green MSP Alison Johnstone supports the car ban call.
She said: “We are in a situation where people are being told not to drive outwith their local authority area and to exercise as close to home as possible and the park sits in the middle of a lot of high density housing, so it's hugely attractive for a lot of people.
"But it's quite disconcerting that you're sharing a park - quite unusually - with vehicles. People have got little children on bikes. Even adult cyclists have told me they find it intimidating cycling through there. It's not the widest road in the world and there are cars in either direction. To really enjoy your park fully you need to make sure that road is closed.”
She said she had written to HES and Culture Secretary Fiona Hyslop in May last year on the issue.
She said Ms Hyslop had replied these were operational issues for that organisation. "But she did say interestingly she was 'supportive of the idea' of proposed road closures.
“Here we are heading into March, the weather is getting better, more people are outside, they want to be able to enjoy their park.
"We need to have a discussion between the local authority, HES and the government, and do everything we can to make sure this is a safe space.
"I think the government needs to get involved and give HES some encouragement."
A government spokesperson said: “The management of Holyrood Park is the responsibility of Historic Environment Scotland and as such, decisions about the park are operational issues for that organisation.”