Wethersfield splits this in a way that catches people out on the first phone call.
Parks and Recreation oversees the development and use of town parks, recreation facilities and athletic fields. The current plan of conservation and development says the Parks group within Public Works maintains the outdoor spaces and athletic fields. So programming and physical grounds maintenance are not necessarily held by the same office, and a goose problem is arguably both.
In practice that means the useful first question is not who owns the park, it is who owns the decision. Whether a program can happen at all tends to sit with the facility and programming side, while what can change about the mowing line or the shoreline sits with grounds.
One more thing that shapes the work here more than it does in a newer town. Old Wethersfield is one of the largest historic districts in the state, and Cove Park sits at the edge of it. Work in and around a district like that is visible, it is watched, and the way a program looks to a passer-by is part of whether it survives its first season. That is an argument for a quiet, scheduled, obviously controlled operation rather than anything that draws a crowd, and it is one reason we plan the timing with the site instead of turning up.
For anything physical near water the wetlands process applies as it does everywhere. Moving birds off maintained turf with a dog is not a regulated activity. Planting a buffer, setting a barrier at a pond edge or changing drainage can be, and that is decided locally rather than by a contractor.
There is one more Wethersfield-specific caution about the water itself. Several of the smaller ponds in the town's inventory do not have their ownership and boundaries established in the public record, so a proposal that assumes they are all town-owned is guessing. That is a parcel-level question and it belongs at the start.