Clarence House was built in 1865, well before cars were invented and has no off street parking
Fortunately, however, parking is free and unrestricted on Park Avenue which is a wide road. It is rarely a problem to find a space very close by at most times of the year although it can get busier during the day at peak times over the summer.
But don't let worries about parking cause you any stress - this isn't like the mean, warden infested streets of the UK'S major cities - even during the popular Ventnor Fringe (centred in the park right opposite) we have always been able to park freely within a couple of minutes walk. Moreover it is always possible to temporarily unload you car from right beside the house as there is a driveway you can partially block for a few minutes.