You can sign up through Apple or through Paramount+ directly. Whichever you prefer, and depending on the devices on which you would like to use this service. (Access on third party devices or on the web requires a subscription with Paramount+ directly.)
Recently expired subscriptions through Apple may still be listed in your subscription overview; you could then renew from there. Subscriptions through Apple that expired longer ago (more than a few months) are not remembered, and you would have to sign up as a ‘new’ subscriber again (but without any perks that a first time subscriber might get).
To sign up through Apple, find your Paramount+ app on any of your Apple devices (iPhone, iPad, iPod touch, Apple TV box, Apple Vision). Then open that app, and sign up for access. Subscriptions started in-app on an Apple device, would be administrated through Apple.
To sign up through Paramount+ directly, go to their website and sign up. You would need Paramount+ specific account then, not your Apple Account.