I have a Clone on the battlefield, not copying anything. (It has a +1/+1 counter let's say.) I also control Parallel Lives. I cast Rite of Replication, targeting the Clone. Can I choose a different creature to copy for each token, or do they have to copy the same thing?

You can choose a different creature for each Clone to copy–they do not have to copy the same creature. The relevant rule here is 616.1f. Parallel Lives' replacement effect applies to the action of creating a token (note that its Oracle text is different than its printed text), while Clone's replacement effect applies to the event of that object entering the battlefield, which is a smaller part of the larger event of creating it.

Since Parallel Lives applies to the larger event, its effect must be applied first; after doing so there will now be multiple independent Clone tokens entering the battlefield, each of which has its own individual replacement effect to apply. Those tokens are all independent of each other, so what you choose for any given one of them doesn't affect any of the others.