Zoning originally came about for a few reasons. One was that industrial pollution was a serious issue and people wanted that away from other types of land use. The other was the rise of the car. Zoning has allowed places to require minimum lot sizes or state that mixed-use isn't allowed. One of the reasons we have so few walkable communities in the U.S. (even small ones like old towns used to be) is that zoning doesn't permit it. Because of this, areas that are walkable and are mixed-use are extremely expensive to live in.
Zoning a required minimum number of parking spots is indeed a terrible thing. Back where I went to college, that was a major headache for every business around campus. A bar was blocked from expanding because it couldn't acquire enough parking spots (and you want people driving to a bar why?), and the public library was required to put in a three story car park across the street when it renovated, demolishing a half a block of shops and apartments. This despite being a block and a half from a seven story car park which was never fully filled.
New York, on the other hand, doesn't seem to give a flying fig about parking. You drive, it's your own problem.
http://www.streetsblog.org/2012/01/06/times-architecture-cri...
If you visit a city like New York, you'll notice that everything is zoned a bit differently; heck, it looks like it hasn't been zoned at all. Most residential buildings have a store of some kind in the first floor. Consequently, in a city with intermingled commercial and residential buildings, you're never more than a block or three from a grocery or a drug store, and generally never more than a block from a coffee shop or a deli.
Doing your grocery shopping via mass transit is insane. That's why in a city where 75% of the population uses mass transit as their primary form of transportation, you don't. You just walk across the the street.