When Robb was alive and after Hoster died, Edmure remained Lord Paramount – but had knelt to acclaim Robb as king after the Battle of the Camps along with the rest of the Riverlords.
The Lord Paramountcy of the Riverlands seems… hmm.
It seems to be less recognized as “legitimate” by a lot of people and especially by the throne compared to the other Lords Paramount. Or at least as something they’re more capable of fucking around with than other Lords Paramount.
Like… the Lannister regime had no trouble whatsoever taking the position away from the Tully’s and handing it to Petyr Baelish, who isn’t even a Riverlander, of all people.
But with the North, they didn’t just declare “the Starks are out, the Boltons in, they’re your Lord Paramount now.” Roose Bolton was declared Warden, but the Starks, theoretically, retain their right to Winterfell and their Lord Paramountcy; its why you need the fiction that Ramsay is marrying Arya, to establish their legitimacy. The Lannister regime also doesn’t appear to be in a big hurry to declare a new Lord Paramount of the Stormlands as well.
I suppose it might have something to do with the fact that all the other Lord Paramounts either come from the lines of kings or (in the case of the Tyrells) are easily able to cloak themselves in the mantle of thousands of years of royal authority. Whereas the Tullys never ruled aught but Riverrun until the Targaryens elevated them, and their control over the Riverlands has always seemed to be more tenuous than that of the other Lords Paramount.
Well, keep in mind that the reason Roose Bolton wasn’t immediately declared Lord Paramount is that Tywin had intended to let Roose Bolton wear himself out fighting the Ironborn and the wildlings, and then get rid of him when he was no longer necessary, to prevent him from challenging Tyrion and Tyrion’s son’s claim to Winterfell and the North. Attainting the Starks wholesale would have rendered Sansa’s hand in marriage worthless, and since the Lannisters had no other way to claim the North, they had to preserve the Starks as rightful heirs.
As for the Stormlands, I think the reason they didn’t declare a new Lord Paramount is that, with Renly dead and Stannis attainted, the Stormlands belonged to Joffrey and then to Tommen as far as King’s Landing was concerned – as the Lannisters don’t seem to give a damn about the customary taboo against holding two great seats at one time.
But it is true, the Tullys are the weakest of the Lords Paramount.