Well, it's not a question that has a definitive answer, but I'll going to base mine primarily over Adrian Goldsworthy's "How Rome Fell".

First, the idea that the Sassanian Empire was a much bigger threat than Parthia is incorrect. They were, after all, the same people, but ruled by different dynasties. The biggest moment of success the Sassanians had (before the last Roman-Persian war in the early 7th century) was right after the dynasty came to power. Therefore, one must wonder how much the armies deployed by them were different than the ones deployed in the Battle of Nisibis, just one or two decades earlier, which was the last set piece battle between Rome and Parthia.

Goldsworthy notes that the Sassanians capaigns that pushed deep into the Roman east were nothing more than large scale raids, meant to solidify the rule of Ardashir I and then Sharpur I. As with Roman Emperors and rulers throughtout history, foreign victories was a sure way to create legitimacy. The success of these campaigns were the outcome primarily of Rome's inability to properly answer to them and the state of unpreparedness the legions were because of civil wars and strife. Duras Europos, for example, a city taken by the Sassanian after a bloody siege, was abandoned soon after because, in Goldsworhy vision, was to far east to be kept.

One aspect of Sassanian warfare that is often noted is their ability in siege warfare, one thing that the Parthians lacked. And indeed, they seems to be quite capable of storming Roman city fortresses. Personally, I wonder if historians didn't tend to underrate the Parthians though. And this is one thing that is also relevant in your question. I'd say the Parthians were more than a relative nuisance. They were certainly not an equal power, just like the Sassanids weren't, but I think the were similar in power, at least during the first decades of Sassanian rule.

All this is concerning only the first decades, and Sassanian Empire lasted for 4 centuries. The frontier however reamined remarkable stable. As Rome power decreased, eventually with the fall of the entire west, both empire became more or less equally powerful. And after their last war (early 7th century), which is the only moment that you can maybe argue the existence of a threat to the Roman Empire survival, or at least its idependence, the Sassanians found themselves in a worst spot, even after fighting most of the war in the offensive, and were conquered by the Arabs in the aftermath.