If you don't like parking you need to start with cars: give people a reasonable alternative. Too many are looking at this from a standpoint of "lets just get rid of parking" - without asking what people will do instead. All too often the answer is they will drive someplace in the suburbs instead where they get free parking.
If you want your downtowns to not have parking you need an alternative. In most cases that means you need to improve your transit in the entire city so people can get there.
It's a chicken and egg problem. If you retain the parking but build transit, people will keep doing what they've always done. Partly because people are resistant to change, partly because the areas you can take transit to are still mostly parking lots.
Places like Los Angeles are grappling with 30+ years of investing in transit with minimal changes in modeshare, because they continued investing in automobility at the same rate.
It takes decades for good transit to change habbits. Others have written about mistakes LA has made on the way, but still transit is making a difference even if small.
I think it depends on what degree of parking we're talking about here, but there are downtowns where the city core is full of large parking lots. That situation makes walking basically impractical because even two adjacent businesses will be a block apart, and everything becomes very spread out. Eliminating most of that parking leads to higher density which is naturally more walkable. The more trips taken on foot, the fewer need to be in a car or other form of transit. Sure this doesn't serve the suburban crowd as well, but it's a city, not an outlet mall.
I think you have to start a step back from cars: the people buying them. At least where I live having a large (and getting larger) 4wd seems to be a source of identity for them, and integral to their lifestyle.
Many cities that have a dense core will have big municipal lots and garages to enable people to park somewhere they can walk from. Of course, the first thing urbanists go after is the presence of these lots in "high-value" real estate.
If you want your downtowns to not have parking you need an alternative. In most cases that means you need to improve your transit in the entire city so people can get there.