I am trying to use the new "Initiatives" features, to allow Epics to have a parent / child relationship to the Initiative level in Jira Cloud, but it appears that Advanced Roadmaps is the only way to implement this and it is not available on Jira Cloud. Is there a way around this?
No, if you want to use Advanced Roadmaps Initiative functionality, then you need to have Advanced Roadmaps. To get it, you need to upgrade to Premium on Cloud.
There are other options - Structure and Big-Picture apps (and some others I believe) can also give you more layers.