The author is not proposing a ban. Just the opposite, he is proposing that we deregulate parking by having people pay for the parking spots they use, rather than forcing someone else to subsidize them.
As far as I know, Russia has no requirements on parking for new construction either. So developers building shopping centres in expensive areas opt to minimize parking space. The result is not clean pedestrian-friendly cities, it's traffic jams and stores going out of business, because as someone else commented here, people will prefer the store with parking, even if it is an extra half an hour away.
I think this problem is better solved by offering functional transit, but it's somewhat chicken and egg. How do you justify spending on transit when everyone drives? How do you not drive when taking the bus multiplies the trip time by a factor of 5?