Unfortunately, no; you cannot deduct (or take a tax credit for) private K- 12 school tuition. Although we could certainly make a compelling public purpose argument that education is always and everywhere a good thing, and therefore should be rewarded in the tax code, the IRS and federal tax law have a different point of view.
The rationale for prohibiting tax benefits for private or parochial school, for grades kindergarten through twelve, is that sending your child to such a school is a voluntary choice you've elected to make, rather than alternatively sending them to a (nominally free) public school. Therefore the government offers no tax breaks for it; even if it is a boarding school, and even if the public school(s) in your local area may be considered less-than-adequate by the child's parents.
Thus, while we can sympathize with for the lack of any tax benefits for private K - 12 education, the law is quite clear on this subject.
That said, a world of new possibilities opens up when someone attends and pays tuition for college. For that, and for much more information on the topic of tax benefits for education, you may wish to download and read IRS Publication 970. A courtesy link to that document is provided here:
https://www.irs.gov/pub/irs-pdf/p970.pdf
Thank you for asking this important question.