It it not yet possible to update a family's pedigree through the REST API. Instead, you must use the save action of the FamilyPedigreeInterface script service, and provide a family identifier, URL encoded pedigree JSON and URL encoded SVG thumbnail.
Note that the code samples generated for you by the PhenoTips documentation will automatically URL encode the pedigree JSON and SVG thumbnail.

Click Try It! to start a request and see the response here!