I'm fairly confident that the parallels are present, intentional, and real. Some portions, however, are about as clear as mud. You'll have to decide for yourself what exactly these parallels are and what they mean. I imagine writing that out would take a Hyperion-length book, and the only true answer would be one by the author.
Thinking about this at all starts with a structural consideration: Hyperion (Simmons) is written in the style of the Canterbury Tales, whereas Hyperion (Keats), although a fragment, seems to be written as a relatively linear tale, dotted with encomia and philosophical considerations.
This difference alone might give reason to be skeptical of a total structural parallel. However, the structure of Simmons' story seems to permit some deeper connection. Simmons' Hyperion follows a pattern where there's a non-narrative sequence of events, and then it delves into a story. Hyperion the poem seems to go down a related path: each book starts with a section that sets the scene, and then people very often tell stories or discover history together.
(Rather than write much of the Keats summary myself, I'm going to quote from this page about Keats' poem, all emphasis mine. I also feel the need to note at this point that it's been a couple years since I read either book or poem in significant detail, and I'd encourage you to draw your own conclusions.)
Book 1 opens in media res with the deposed god Saturn mourning the loss of his empire to Jupiter. Thea, another of the Titans and sister/wife to Hyperion, attempts to comfort him, but, also despairing, she can only weep at his feet. Saturn rouses himself to renewed resistance and is led by Thea to where the other fallen Titans are assembled. The second part of Book 1 focuses on Hyperion, still unvanquished and defiant but apprehensive and suffering from premonitions of death and disaster, preparing to go to the aid of Saturn.
Simmons' prologue seems to take something of a different turn than the poem. It opens with the Consul receiving a message from Meina Gladstone. Unless we're supposed to view the Consul as connected to a deposed Saturn, and Meina as the sister/wife to Hyperion, I find this to be an unlikely comparison. While under some lines of thought Meina could be analogous to a Titan, the Consul is certainly not portrayed in the manner of a deposed god. I'd suggest that the prologue isn't intended to make a parallel to Keats' poem.
The first chapter, however, follows exactly what's happening in that section. The Consul wakes up from cryogenic sleep. Het Masteen is there, and informs him that the other pilgrims are aboard the Yggdrasil. I'm not sure how that could possibly be a closer parallel.
The first "chapter" of Simmons' novel does also appear to actually end "[focusing on Hyperion]," in a tone "apprehensive and suffering from premonitions of death and disaster." Recall that Hoyt's story ends with a discussion of continuous resurrection and crucifixion on a Tesla tree - something that is, in and of itself, foreshadowing. This suggests that maybe we're not supposed to consider the prologue in the story as part of the parallel to Keats' Hyperion.
So at this point, I'd consider the question of "who is Saturn, and who are the Titans?" to be somewhat unanswered. But the possibility for a parallel in the generic is there. I also think it's more likely that no single specific person or group was intended to represent any specific person or figure in the poem, and that facets of the Pilgrims and facets of the people in the stories told are supposed to present comparative ideas only.
The second chapter of the story presents a much more clear parallel:
Book 2 describes the council of the deposed Titans, as they attempt to come to terms with their new powerlessness...
Much of the opening of chapter 2 of Simmons' Hyperion describes the pilgrims' journey through the capital city named Keats (which is definitely a total coincidence with no meaning). The Consul encounters Theo Lane, and has a discussion about how the city of Keats has grown and changed.
The speech given by Oceanus is the most positive; he urges his fellow Titans to come to terms with change as an inevitable part of the natual process, and ends by praising the beauty of his own supplanter, the new god of the seas, Neptune.
Structurally, these conversations are very similar. Lane is pushing the Consul not to go on the pilgrimage, and the Consul rejects it, describing the pilgrimage in a very similar way.
Kassad spends the rest of the chapter recounting his story; this seems to mirror the rest of Book 2 of the poem:
Her lament declares the uselessness of philosophical arguments in dispelling grief, but also comforts Oceanus's wisdom when she describes the beautiful music with which the earth greeted the arrival of Apollo, the new god of song. Enceladus does not agree and urges them all to challenge the enemy, reminding them that Hyperion is still unfallen. Hyperion himself arrives, but he has now accepted defeat.
This honestly exactly mirrors the pacing of the story. Kassad's story starts out as a militaristic reminder that philosophical arguments won't really help that much in the face of this kind of struggle. It then sidetracks into the arrival of Moneta, and his relationship to/with her, in a way that could only be uncomfortably described as pretty close to divine and divine intervention.
Besides, Kassad's story ends after the Shrike arrives.
Book 3 gets a little less clear again, but I think the parallel is still there.
The brief, fragmentary Book 3 describes the valley where Apollo, who will be the new sun god, is coming into his powers. Mnemosyne, herself a Titan, the goddess of memory and mother of the nine Muses, presides over the initiation ceremony. Apollo, whose poetic associations are emphasized by his possession of a lyre, reads a 'wondrous lesson' (book 3, line 112) in her face, and 'Knowledge enormous' (Book 3, line 113) transforms him into a god. Book 3 breaks off with his assumption of godhead.
Simmons' chapter 3 opens with the destruction of Naiad (also definitely not a meaningful name worth thinking about), being burned to ashes. It's a little confusing, and this is where the parallel starts to break down. Coincidentally, this is also where the poem by Keats ends. The closest parallel here has to do with "the nine Muses" in Sad King Billy's City of Poets. Silenus in this appears to mirror the path of Apollo: he writes endlessly.
The ending, however, appears to be more of a meta-point: Silenus' tale ends with the destruction of the Cantos, and he wishes to return only to complete his unfinished work. This is plainly a reference to Hyperion itself going unfinished.
Does the pattern go on past here? I'm not sure. It's hard to say, what with the poem itself being unfinished, and it being so many years since I've read either in much detail. It does leave open a pretty important question: what about the remaining three and a half chapters?
But I think this definitively answers the questions: there is some kind of parallel here, though the book as a whole likely doesn't end when the poem does. The exact way to interpret this might just not be clear.