Is it possible to get Parts to sync to start on the next bar when switching between them?
As far as I know, no it isn’t. However if you choose a part on a pattern - the pattern remembers your last used part. So for example you could select Part 1 on Pattern 1, Part 2 on Pattern 2, 3 on 3 etc. When you change to those patterns the part you previously used on it will be active.