The way I interpreted the policy, if a DVC member is staying in a DVC room, regardless of how it is booked or paid for, parking is free. I would recommend having your DVC card available at check-in to show member status for free parking.
From the Disney website:
Disney Vacation Club Members
Members will not be charged for standard overnight self-parking when staying at a DVC Deluxe Villa, regardless of whether they use vacation points or another form of payment. Members also will not be charged to park when using vacation points to stay at a Walt Disney World Resort hotel.
To me, paying cash is another form of payment. The site says nothing about needing to book through DVC to get free parking, just stay in a DVC villa.