Birding at National Park Zapata is outstanding. We did this guided tour twice with different guides at different months. A car is necessary to go to this three different places.
The best time of day for birding is the morning. Don't go too early, otherwise mosquito might eat you up.
The best months are November until April in the dry season. Rainy season from July until October is not recommended, as areas can be flooded. Shoulder season May and June, showers may occur but easier to find a casa without booking in advance. July and August are the Cuban school holiday.