Unwinding decades of bad parking policy is more complicated than simply ending off-street parking requirements, although that's probably the most important part.
Another key aspect is changing the way we deal with on-street parking. There are currently a number of approaches in Austin, and some of them are better than others.
Free street parking. That's what exists in neighborhoods such as my own, where parking is not in high demand because there are not any popular commercial attractions nearby.
Resident-only street parking. Then there are areas where street parking is restricted to residents of that street or neighborhood, at least during certain times. In Austin this is made possible through the Residential Permit Parking Program, which residents of a street can petition the city to implement if a majority of their neighbors are in favor. This is often done on streets near entertainment districts: South Congress, S. Lamar, Manor, etc.
Resident-only street parking is bad for a number of reasons.