If I fell at a city park and dislocated my knee cap, can I sue the city for medical bills?

It is very unlikely that you could successfully sue:

1) When you use a park, you assume the normal risks attendent upon the activities (e.g. sports, running, hiking, etc.) you engage in. A normal risk is falling and being injured. Thus, unless there was something extraordinary or unusual which caused your fall--for example, a poorly maintened staircase, which cracked under you--you would most likely be held to have assumed the risk (and be unable to recover compensation for) the fall.

2) While you can check the municipal code to see what, if anything, it says about padding, state law more generally does  not, I believe, specify any minimum level. Given that parks also include concrete, pavement, rocks, tree roots, hardtru, etc., I suspect strongly that a minimum level of padding is not required, but rather, matters are judged by whether there was negligence on the park's part. 2" of beauty bark padding would very likely not be considered to be negligent.