From what I understand, David Deutsch invented the idea of quantum computer as a way to test Parallel Universes. And later people went on and built the quantum computer. Are you saying that the implementation of a quantum computer does not require any kind of assumption on computations being run in parallel universes?
That's right, it doesn't. The implementation of a quantum computer does not prove or disprove the existence of parallel universes.
In short, no.
It's just not how it works. All this type of quantum computer can do is test some of the more dubious objective collapse theories. Those are wrong anyway, so all theories that are still in the running agree.