It should be very frustrating to argue with people who complain too much about political dynasties. You can't avoid them but you can minimize their impact. The only problem of political dynasties is when incompetency succeeds incompetency or when everyone in the family is free-riding thanks to the popularity system. So how would a parliamentary system guarantee keeping political dynasties in check?
It's stupid to keep complaining about political dynasties while loving the Aquinos. Okay, we only have Senator Bum Aquino right now but don't tell me it wasn't a political dynasty when the late Cory Aquino took over her husband or when Noynoy Aquino becomes president just like his late mother. For the nth time political dynasties are bad only if they don't deliver. The problem with political dynasties in the Philippines is that they fail to deliver due to a popularity-based system. One member of a clan runs for office and everybody follows regardless whether or not they are competent or incompetent.
How would a parliamentary system reduce the power of political dynasties? It would be because just because you're a member of a popular clan wouldn't guarantee you a seat in power. I would agree that both the Dutertes and the Marcoses have to be placed in check like every other family. It would be good for Dutertes and Marcoses to have a political dynasty if everyone involved is doing a good job. But the problem will come when stupid members of the Dutertes and Marcoses will pop like popcorn because of the popularity of President Rodrigo R. Duterte and Bongbong Marcos. If wayward members of their clan should choose to run a competency-based system won't allow them.
The point of keeping political dynasties in check is that only competent members of a certain political clan will be allowed. None of the incompetent members can take a free ride just because a competent member is in office. Competency-based systems could care less if the person is related to a competent politician. If he or she can't prove himself then he or she won't get that position. There will be rules that will put people to prove themselves. It wouldn't put a person into power just because of their family background. It's their individual competency that's being checked. This means political dynasties here will only exist if other members of the family are competent people.