Vonage's Voice API allows developers to connect multiple parties to the same call by using a webhook endpoint and a conversation action in the Nexmo Call Control Object (NCCO). This is achieved by creating a Flask application that defines a webhook endpoint, making it accessible over the public Internet through tools like ngrok. The developer needs to create a Vonage Voice API Application, link it to their virtual phone number, and configure security information before initiating a conference call. The process involves purchasing a virtual phone number, installing dependencies, setting up the Flask application, and running the Vonage CLI commands to create an application and link it to the number.