BART now allows Clipper cards for parking like EZ Rider. First, you need to create an EZ Rider account with BART  and link the parking account (credit card is required) with your Clipper card (by entering your Clipper card serial number). BART will then send you a parking hang tag. You can park at a BART station showing the hang tag and tap your Clipper card at the parking validation reader. You don't need to pay cash nor have to remember your parking stall number.
Payment for parking comes from your credit card registered with BART and not from the Clipper Card. The Clipper Card is used for fare payment and parking validation only.