Are platforms such as Airbnb and Eventbrite considered Payment Service Providers (PSPs)?

It is my understanding that they are not. From what little I know about these companies, they presumably use a licensed payment provider, who either themselves or via an upstream processor/banking agreement have taken care of the money transmitter issue as well, i.e. their upstream provider is licensed for money service business / money transmitter legal requirements.

They would still have to adhere to the payment processing rules as setup by their processing partners.