Can I automatically apply an object style based on paragraph style?

That's pretty much all you would need:

BEFORE:

AFTER:

Yellow represents applied Object Style.

1st Task - should be run using Play button at the top - will load info about all ParaStyles that you want to include - there is only one rule for that, the one in the middle - but you can add as many as you need - to build the list of elements to process - steps:

- Clear List -> Texts - just in case there are no leftovers,

- Load Text Search - basic version of the Ctrl+F / Find/Change dialog,

- Mark as... - selects all elements on the specified list for further processing.

2nd Task - in full version, should be run using Rewind button, located in the middle - steps:

- Modify current selection -> Parent TextFrame - switches current selection from found text to TextFrame.

- Apply Object Style

In free version BatchMode is not available, so you can't use 2nd Task to process all elements on the bottom list - found results - automatically - but you can simulate BatchMode by double clicking on any cell in TYPE column - in this case "Found" text - which will  select object/text on the list - then you can select 2nd Task and click Play at the top to execute 2nd Task.

In short - BatchMode is doing this double clicking for you.