The council has earmarked parks as a service which it wants to become self-funding in its 2017-22 Corporate Strategy and said it wants to "develop new models of community asset management and leadership of key neighbourhood assets, including libraries, customer service points, community buildings, parks and green spaces."
The plan states that in this financial year it would work "with stakeholders to develop a robust plan to consider ways of raising income and reducing cost to create a self-funding service".
Cuts outlined in its current budget, passed on February 21, include losing the entire 2016/17 parks budget of £3.9 million by cutting £425,000 in 2017/18, £632,000 in 2018/19 and £2.86m in 2019/20. The department brought in income of less than £1m in the financial year just completed.
In the petition, the forum states: "Bristol Parks Forum believes this is impossible to achieve in such a short timescale. Many other big cities are exploring this option, some have already decided it simply won't work; this was also the conclusion of last year's Select Committee of the House of Commons.
"We, the undersigned, call on Bristol City Council to withdraw this budget plan and work with the Parks Forum and others to develop a realistic alternative."
Forum members, made up of dozens of park friends groups and other community groups, say they are concerned about the "high risk that there will be a drastic cut in the maintenance of parks that will lead to a downward spiral of reduced use and increased anti-social behaviour as standards fall."
They say they just do not believe it can be possible to more than quadruple income in such a short space of time, and would "require significant investment of both of capital and commercial expertise over several years.
"Bristol Parks Forum believes that while a significant increase in income can be achieved; having well maintained parks operating on a cost neutral basis by 2019 is simply not possible."