closer alignment between local/parallel and Fluent-based parametric

These have been developed independently. Look into how they could be brought closer together.

Start by comparing the local/parallel implementation with the Fluent-based parametric implementation named in the issue. Identify where their interfaces and behavior differ, then determine a concrete alignment plan; the issue provides no files, tests, or completion criteria, so these must be established before implementation.

Written by the indexing model from the issue text.