Parriwi is the best way. Military/Spit Road is like a river during peak hours (takes ages to cross), so you are better off keeping to one side of it unless there is a pressing reason not to (like more hills & laziness :)).
Bridge-Rocks-Quay-up the elevator to the Cahill path then across the Botanic Gardens from the park maintenance depot (verboten but doable if respectfully slow and sneaky observant for rangers in golf buggies)
Bridge St or Hunter to the Pink Palace then across the Domain (legal)
Bridge or Hunter then down the expressway in the breakdown lane, left at the bottom of the hill into Cowper Wharf Rdy.
Down the back of the Art Gallery and elevator to Bourke St.
Whatever you fancy to St Mary's Rd, left onto Sir John Young Crescent and rt on to Cowper Wharf Rdy (this is my usual take-the-lane route)