Do I Need to Purchase The Events Calendar Pro as Well?

No, you do not need The Events Calendar Pro to use Registrations for the Events Calendar Pro.

The Events Calendar by Modern Tribe is a widely-used plugin for adding events to your WordPress site. Registrations for the Events Calendar is our plugin which builds on The Events Calendar by allowing attendees to register for your events.

Each of these plugins has its own pro version which adds useful features for a yearly fee, but these pro versions are entirely independent.
Both the free and pro versions of Registrations for The Events Calendar are built to work with either version of The Events Calendar.