Because having parking isn't what makes it car centric. Having bike lanes doesn't stop it from being car centric. Cities and neighborhoods are designed for cars, and cars will always be the preferred mode of transportation as long as cities and neighborhoods are developed that way. Cities need parking because they are car centric. Cycling, and living in a city with cycling, is a luxury. It's not a bad thing, but it's not going to help, either.
To answer your question, public transit is how you get around without a car. Spend the money on infrastructure, and reserve lanes for busses and light rail. Reclaim roads entirely as pedestrian paths. Force developers and city planners to create walkable communities.
Revisiting the vegan metaphor, everyone agrees that beef production is bad for the environment. If you're running a steakhouse, you've built your entire restaurant around beef. Adding a page of salads to the menu is nice, but it's no less of a steakhouse. They won't sell significantly fewer steaks just because there's a salad on the menu. People will still choose the steak, because it's a steakhouse and that's where people go for that specific thing. To reduce the amount of consumed beef, you have to change the restaurant.