If only Austin had decent public transportation. The new rule may just make it harder to find parking but not a lot else on the car side of the equation.
But waiting until there's good public transportation before you do anything to change car dominance just maintains the status quo of shitty public transportation... Abolishing parking minimums won't mean car parking disappearing overnight, but it will open up more development opportunities (whether residential, commercial, or mixed use) in neighborhoods that are already near transit or other walkable areas.
Austin, like almost every city in the US, has a ridiculous excess of parking spaces - people only complain about parking because they have an unreasonable expectation that their parking space should always be free and right in front of their destination. The only real negative consequence for drivers of abolishing parking minimums is that some people may start needing to walk slightly further from their parking space to their final destination. The benefits that come from that small cost are immense, and abolishing parking minimums is an absolutely crucial early step for any city trying to initiate a long-term transition towards becoming a more eco-friendly, livable, walkable city.