I've inherited a Parse codebase in the past, and thought it was pretty gross. You're stuck building with their ORM abstractions, so it hurts when you need to do something complex. I view Parse as Facebook abandonware that's been salvaged by a community of developers who need to maintain legacy apps on life support. I think it'd be unwise to start a new project backed by Parse.
Having to deal with one-size-fits-all database "adaptors" for either Mongo-flavored or Postgres-flavored Parse is limiting, whereas having direct access to Postgres is a huge benefit of Supabase.
With Supabase, you can leverage all of the goodies that Postgres has to offer, which is exactly what I want as a developer. Even something as simple as writing custom SQL is a well-defined operation within the Supabase ecosystem, whereas with Parse you're in undefined glhf territory. Add that to first-class support for realtime updates, row-level security, and their other features, and Supabase comes out way ahead in my opinion.