WordPress 4.9.4 does not support the
post_parent attribute in the REST API, so I wrote a plugin that does.
Download here: https://github.com/csalzano/wp-api-add-post-parent
Someday, WordPress core will allow the manipulation of the
post_parent attribute, and I look forward to deleting this plugin and this blog post when they become obsolete. Until then, I believe this implementation resembles how core will handle this.