We watch the parade from the Historic District at various points around the Battery (White Point Gardens). It does start in Mt Pleasant, crosses the Cooper/Wando rivers and comes down the Charleston peninsula, turns up the Ashley River at the Battery and travels along Murray Blvd and on up to the end. This is where you will have a closer view of the boats. The historic homes along East Bay St, the high and low Battery and (not as historic) Murray Blvd are at their finest on this night. Many are holding parties and you might see groups of people in black tie and evening gowns at one home, casual at another, moving about through open windows and out on piazzas. Some party goers from the homes come out and stand by the river when the boats pass by. It's a very festive night. There will be fewer people on Murray Blvd and easier parking but it will be at the end of the parade. If it's really cold or you have someone who can't stand for a long time you can sit in your car on Murray parked at the waterside and hop out when the boats get there. Or you could also take a lawn chair to that area as the sea wall is lower.
If you want to be among the crowd, lots of people watch from Waterfront Park off Vendue Range. There are several parking garages nearby on East Bay Street and also on Cumberland Street. I am not familiar with what goes on in Mt Pleasant. Hope you enjoy!