If the park is a DIY, then I think the recommendations above are not half bad. If it is a city sanctioned skate park, I recommend getting the city involved in some way, shape, or form. It is a public health issue. It might require some fundraising from the community though.
Ideally, in the case it is a city sanctioned park, they'd modify the existing structures to prevent the pigeons from returning and then just power wash/pressure wash the bejeezus out of everything.
EDIT: It could be a worthwhile story to pitch to the local newspaper to bring attention to the issue if the city are being jerks.