There are several ways of interpreting the situation. But you're right, they are jumping between parallel worlds. However, to get to those parallel worlds, they must travel back in time.
Let's put it this way: The Watanagashi Festival always takes place on 19th June 1983. It never takes place on any other date. So, in Onikakushi-hen, Rika continues living until 19th June. However, everyone dies and she must travel to a different kakera. If she simply jumped to Watanagashi-hen at that point, it would have already been past 19th June, and so everyone would already be dead, which is really no good for Rika. Therefore, she must travel back in time to the start of June, and enter the kakera at that point.
To be perfectly honest, there simply isn't enough detail to be sure of how Rika's jumping between kakera actually works. However, the world of Higurashi seems to conform to the
many-worlds interpretation. In which case, all branches must converge at some point in the past. You could say that every single branch can be traced back to a single tree trunk, which would be the beginning of time itself. Everything that ever happens simply branches out from that.