What’s odd here is that you shouldn’t need anything special to get a page break after a part page. You don’t have to try to find a way to insert a second break after it, because the chapter break itself should do that. It would take additional customisation in fact to arrive at a scenario where the part title and chapter title occupy the same page.
You can see the stock behaviour demonstrated in the built-in “Paperback” compile Format. If you assign your parts to “Part Heading” and chapters to “Chapter Heading”, then it should work without any tweaking. (That Format will keep all breaks on the recto page, so you may see extra blank pages if anything.)
In fact there is just such a setting, and maybe it is already set to something, which would explain your result. The optional Override separator after setting is what you are looking for—and you would want to be checking the Separators for your parts here. If you turned that on and then left it as “Empty line” say, then it would in fact override the page break of the chapter Layout that follows it, and lead to both items being on the same page.
Case in point, this is how the default Manuscript formats work—where Parts are printed on a line above the first chapter of the part, no break between them, while chapters otherwise ordinarily break to a new page elsewhere in the manuscript, just like you describe.
All of which is to say, Scrivener can do both the thing you are describing as happening, and the thing you want. So I’d take a look at those two stock examples and see if the format you’re working on resembles Manuscript more than the former.
If the Override separator after setting is enabled, simply switch it off. Like I said above, the chapter break itself is sufficient to start a new page, so long as nothing else messing with it, like this setting.
P.S. Separators between are described on page 623:
So for that to work you need two adjacent binder items using the same Layout. Classic case of that happening are scenes, where something would be inserted between scene files, but not before them nor after them. So that setting will not work with something so disparately placed as parts.