Sullivan said Johnson needs to spell out what he would do with the parks and what can he do better than the Department of Culture, Media & Sport (DCMS), which currently funds and runs the parks.
"If the mayor is serious he has to put out a business proposal," said Sullivan. "The DCMS needs to invite the mayor to make public proposals. Royal Parks could make its own proposals too."
Sullivan suggested the DCMS may give the mayor a "dowry" to run the parks and help make them self-funding in the same way as green spaces in Milton Keynes. He added: "There are real opportunities for Royal Parks to say some things are so valuable and so expensive to run they should be charged for. They need to consider recouping some cost by charging. Sections of the community couldn't afford that so there should be a discount.
"There's an enormous number of tourists and cost of upkeep is astronomical. In certain instances charging tourists is a possibility. Certain buildings could turn into pay-for-heritage buildings around Kensington Gardens, for instance.
"There could be a Royal Parks exhibition of their past - tourists love that sort of thing and would pay to see it. But I don't think charging to get into parks is acceptable."
A mayor's representative said: "There are many synergies with the mayor's other responsibilities, including the environment, tourism, transport and culture. At present the mayor has no jurisdiction over Royal Parks, but there are detailed conversations taking place."
Meanwhile, Johnson along with the London Borough of Richmond has called for plans to charge for car parking in the Royal Parks' Richmond Park to be scrapped.
- Around 30 Victorian and Edwardian fountains are to be restored in the Royal Parks by 2011. The Tiffany & Co Foundation will fund the £850,000 restoration.