My friend, George, met you last week in Cuenca. We are considering moving there soon, but are concerned about our dogs.
You mentioned you brought your dog. We like the old Colonial area, but is there a place to have (walk) dogs and play areas for them? We want to avoid the gated community scene.
Depending on where in colonial section of Cuenca that you'll live, you could head to the river (Río Tomebamba), which is a large green space, or up to the Heroes de Verdeloma area, which has a bit of greenspace.
Or go to one of the plazas that dot the downtown area. Many have a bit of grass. Just be sure to pick up after the dog, because the neighbourhood kids play in the parks.
If you are coming to Cuenca with your dogs, it´s better to rent something with greenspace... there are a lot of nice places very near from the downtown. A friend of mine owns a little warm house 25 minutes from the city and it´s for rent right now. If you are interested just ask :)