I think the answer to that is no and the reason involves entropy. Of all the possible universes most have very low probability, so when we "create" a bunch of parallel universes for our computer, those universes could already be existing which would interfere with the calculating (not in the sense of slowing it down because of too many things running on it, more like a radio channel with lots of interference from nearby channels), but this won't happen in practice as these universes will have very low corresponding probability.
On the other hand there is self interference, from poor engineering of the computer. I think this is a very significant problem now - but it will usually be described in terms of decoherence.