i can't speak for the buses that pick up outside the district, but any bus locale (China town/mcpherson sq./dupont/logan circle) in the city won't have free parking with the exception of sunday. You could possibly park in a gargage downtown (in the mcpherson sq. area anyway) but you'll likely pay enough to not make it worth while.
Just metro to the bus pick-up site.
Also, mega bus is only $1-2 bucks for the first few tickets. After that it'll hop up quickly but it's still a nicer deal than the china town buses. Those stress me out, i always feel like I have to start throwing elbows to guarantee a seat.
