I had a lovely short break at Warwick House - a very comfortable small B&B with lovely owners and excellent cooked-to-order breakfast. Right by the Jubilee Pool in Regent Terrace, and it even has a small car park. Highly recommended.
http://www.warwickhousepenzance.co.uk/
There are a bunch of B&Bs on the same small street which all look nice, so worth exploring Google Maps for their names then checking out reviews. It’s a great location.
We plan to rent a car for a day trip, but shouldn't need parking where we're staying.
Also: I meant to add - Penzance is one of the (few?) places in Cornwall where you can have a great trip without a car. There’s a regular bus that goes all round West Cornwall, including St Ives, Lands End and Porthcurno (for the Minack Theatre). You can buy a day ticket and hop on and off. You can also get to St Ives by train (I think you change at St Erth). And there are local buses from Penzance that run west (to lovely Mousehole) and east (to Marazion for St Michael’s Mount). These two are also quite simple walks as well.