In case of Parse Server, a well indexed Mongo will perform better since when Parse was not open source, the database behind the service was MongDB (with a special storage engine).
The code base , and architecture is designed for Mongo ( and specialy NoSQL database). Postgres was added because in 2016, noSQL databases were not popular as today. And postgres database services were cheaper than Mongo services also.
Another suggestion, Mongo is the preferred database, some features are not well covered or fully supported at this time on postgres, some bugs can exists on postgres and not on mongo.