Giving parents choice would mean the parents could find a school that works for their child. It would also create more competition between schools as schools strive to gain more students, this would force schools to pick up their game and offer better educational standards. It works here in Australia.
I think if parents really care about their child's education, they should have the option. There are many cases, where parents can't afford to live in a certain area, but their kid shows great promise. That kid should be able to go to a better school that can challenge them, in a good way. Some parents just can't afford private school but would love to send their kid to one of the "better" public schools. I think the kids would benefit from it.
Having grown up in a county with a number of high schools with different focuses (information technology, IB programs, biology, arts, etc), I appreciated having a choice when it came to my education. I was able to go to a school that challenged and interested me, making me much more engaged than I would have been at a standard high school, and putting me ahead of the curve when I began an engineering curriculum at my university.
Parents should be allowed to send their children to whatever public school they want, so long as they pay for it. If a parent wants to have their kid go elsewhere, they should have to pay the same level of taxes that someone in that other community has to, it's only fair.
I think parents should not be allowed to send their children to public schools outside of their district simply because it would overcrowd some schools more than others. I think it would make things very complicated, it would overcrowd "preferred" schools and could leave children that actually live in the "preferred school district" to have to commute to other schools.