Parent site link on sub site

Hello Carl_Williams 

not really, the modern site concept based on sites (=SiteCollection), not subsites.

Here you can find more information: https://learn.microsoft.com/en-us/sharepoint/information-architecture-modern-experience 

From this article:

Classic SharePoint architecture is typically built using a hierarchical system of site collections and subsites, with inherited navigation, permissions, and site designs. Once built, this structure can be inflexible and difficult to maintain. In the modern SharePoint experience, subsites aren't recommended. 

Best, Dave