New York has set such a policy in place and it's been upheld through multiple trials and appeals. Perhaps once it's settled in the higher courts it will become standard policy in a lot of other places. Once can hope. http://jurist.org/paperchase/2015/01/federal-appeals-court-u...
It seems to me like New York's current law is the same as California. From the article you linked: The state law provides two exemptions from the mandate, ... the second for "children whose parent, parents, or guardian hold genuine and sincere religious beliefs which are contrary to the practices herein required. The NPR article seems to discuss a father specifically attacking the personal belief exemption in Califor…
Before the obvious objection, there are plenty of examples of people who try to create a special purpose religion to get around some or other rule, and only a few cases where it has actually worked.
Also as the top level poster alludes to the second circuit recently ruled that the religious exemption is not constitutionally required. Phillips v City of New York (http://caselaw.findlaw.com/us-2nd-circuit/1688862.html).