However, I think this is just between two separate staves that aren't grouped. Is there a way to separate between staves in a part?
You are correct, that only affects the distance between staves within the same system. For inter-system spacing, the setting you're looking for is in the \paper block. If you look at the NR section on the \paper block, in the "Flexible vertical spacing \paper variables" subsection, and scroll down a bit, you'll see a list of different variables that control how all the different kinds of elements are spaced relative to each other on the page, depending on what is there. In your case, the variable you're looking for is 'system-system-spacing (i.e., the distance between two systems in the same score), which you'd set using the same properties (e.g., basic-distance, padding, etc.) like this:
system-system-spacing.basic-distance = #10
Of course, adjust to taste.
For your reference, the full default settings are: