The next step is to establish causality. They will have to deliberately remove (and maintain removal of) this bacteria from certain people and measure the progress of parkinsons over many years. There will also be a control group given placebos. The difference between these two groups will be the factor that crystallizes the claim made here.
What claim would that be? The only claims made in the article was that a) it's a breakthrough, and b) "Researchers say certain strains of gut bacteria are the likely cause of Parkinson's disease."
The article also refers to the bacterias as "probable causes".
Everything you pointed out supports these observations.
While technically association is a prerequisite for causation (correlation doesn't imply causation but causation implies correlation) probable "cause" is too strong of a term here because the experiment didn't actually do a causative test. "Possible cause" might be better here because the title heavily implies a causative study was done.
Do note that in science nothing can be proven so in actuality probable cause is really the highest form of verification that can be made. That is the claim in the title.
in English, "stronger claim" has a meaning ambiguous between "the claim is strongly defendable, i.e. more likely to be true", and "what is being claimed is of more powerful effect". I think here the claim they are trying to point out is the powerful effect, to say "this should be pursued, its importance is elevated by its potential strong causal/explanatory effect"