Quick update.
I contacted the city council and was given the name of the guy in charge of Parks and Recreation. I emailed him, and he told me to come along to a Parks & Rec Board meeting - held once a month, open to the public.
So I went along last Thursday, with a handout I'd prepared that basically put forward the proposal - a low-cost proposal to revitalize an underutilized park in an under-served corner of [city]. I included a one-sentence summary, and bullet points on WHAT [phased installation of outdoor exercise equipment], WHERE [a certain park, opposite a mental health home so some concern in community in past at amount of smoking that goes on there - accordingly not a great location for playground], WHY [to encourage fitness, because we don't have anything nearby, to attract adults to the park (the mental health facility residents use the park a lot which is great, but it would be great to have other residents using it too] thus increasing safety/sense of community, inclusivity (something the care residents can use, and all other residents regardless of income]. Also mentioned estimated costs, and included photos of the park and proposed equipment.
The board said they really liked the idea, they agree the park is underused and it is scheduled for capital improvements in 2016, and they agree that exercise equipment would be a good use of that earmarked money. So I will keep following-up, and be sure to be at the meeting where they decide what to spend the money on so I can remind them of my idea :)