In a city this size the answer changes by a few hundred yards, and getting it wrong wastes a month.
Parks, Recreation, Youth and Community Services is the first stop for city ball fields and parks, and it has a Board of Park Commissioners alongside it. A problem at Rockwell Park, Page Park, Hoppers Birge Pond or a ball field starts there.
Public Works is a separate structure with its own divisions, and its Engineering function carries stormwater. That is the door a drainage or water-quality question goes through, and given that the city's own stormwater work names waterfowl management, it is also the door that already has this subject on its desk.
The Water and Sewer Department protects the municipal watershed lands, which are a different category again from a park with a pond in it. Reservoir and protected watershed land is not something a contractor walks onto because a flock happens to be there.
For anything that would change ground rather than move birds, Bristol's Conservation Commission serves as the city's Inland Wetlands and Watercourses Agency. It adopts and administers the wetlands regulations, decides regulated-activity applications, changes mapped boundaries, enforces permits and determines exemptions. Its application requirements are detailed, including owner consent where needed, a description of the affected wetland or watercourse, proposed disturbance, soils and vegetation, alternatives, a site plan, abutter names and a stormwater certification for a 50 year storm.
Two things follow from that. Shoreline planting or a barrier at a pond edge is a filing, not a favor, and moving birds off maintained turf with a dog is a different activity that should not be confused with it. Bristol publishes meeting and filing deadlines but no single universal processing time, so treat any promised approval date as invented.