-- Mark Tatz and Jody Kent (1978), _Rebirth: The Tibetan Game of Liberation_: https://www.himalayanart.org/search/set.cfm?setID=1524
-- May-Ying Mary Ngai (2011), "From entertainment to enlightenment : a study on a cross-cultural religious board game with emphasis on the Table of Buddha Selection designed by Ouyi Zhixu of the late Ming Dynasty": https://open.library.ubc.ca/soa/cIRcle/collections/ubctheses...
-- Jingyi Yuan (2021), "Blurring the Boundary between Play and Ritual: Sugoroku Boards as Portable Cosmos in Japanese Religion": https://etd.ohiolink.edu/acprod/odb_etd/etd/r/1501/10?clear=...
In Europe, many parlor games had some moral, didactic, or personal aspect to them:
-- Innocenzo Ringhieri (1551), _Cento giuochi liberali, et d'ingegno_ gave players 100 different games that were kind of complicated and often didactic in themselves, but the idea in making them complicated was for players to make mistakes they'd pay for at the end by answering questions that often had a moral dimension. I've made a rough translation here of seven of them that were translated into French in 1555 and that have to do with classical mythology: https://wobbupalooza.neocities.org/1555
-- One specifically political parlor game that I can recall off the top of my head is the variant of Avocat that GutsMuths gave in 1796 called Parlament, in which players become King, Chancellor, Secretary, etc. to address strategic matters of state such as "whether one should be allied with the [Ottoman] Porte, recognize the [French] Republic, continue the war, [or] levy a new tax or not," the trick being to respond to the question in lieu of a partner: https://www.google.com/books/edition/Spiele_zur_Übung_und_Er...
-- But that's pretty close in time to the 1804 astronomy board game that Margaret Bryan consulted on: https://collections.vam.ac.uk/item/O26305/science-in-sport-o... / https://www.rmg.co.uk/collections/objects/rmgc-object-264736 / https://www.cultureandcosmos.org/pdfs/16/Saridakis_INSAPVII_...