You can park down by the harbour there but can be quite busy and tight in there at times. Definately worth considering driving to Rock and parking there and getting the ferry across, though depending on the tide it may mean you taking a bit of a short walk into Padstow. If you are the energetic sort try driving into Wadebridge and hiring bikes then following the Camel Trail to Padstow. The cycle places in Wadebridge let you park there for the day.
We holidayed near Padstow in August a couple of years ago. There is a large car park next to the lobster museum. If you get there before 10 am you should be able to park quite easily. There is also a large car park at the top of the hill just before you go down the hill into Padstow.