No, not city zoning requirements. In Madison, there are no (car) parking requirements in the downtown area. If someone wants to build an apartment building or offices with no parking, they are free to do so.
But I heard a strange story last week, and this article echoes the same problem. A woman wanted to open a store on the Capital Square, but couldn't get a business loan. Why not? Because she had no parking. She wasn't worried about it. There are parking ramps quite close, and thousands of people would pass by her store by bus, foot, or bike every day. But the bank wouldn't give her a loan. No parking, no loan.
But as soon as the city turned the outer lane of the Square into a parking lane, viola! she got her loan. Doesn't matter how the majority of her customers got the the store. Doesn't matter that people get on and off the bus yards away. Doesn't matter if 10,000 people walk by during the Farmers' Market, or bike by as they come and go during the week. No parking in front, no loan.
What this means is that walkable, bikeable, transit friendly areas of the city are being penalized for being accessible and not car-oriented. If one small store couldn't get a loan, think what the restrictions would be for a larger building! [For more on how urban, mixed-use development is being screwed by federal loan rules, see this recent Forbes article.]
Now, most developers of larger buildings will build parking anyway, regardless of how expensive it is, because they think they can't rent out their space - residential, retail, or commercial - without parking on site. But if a brave soul decided that downtown Madison already had enough parking, and more office (or retail, or residential) space was a better thing to build than parking, they would be shut out of loans.
How crazy is that?