It is possible for today to attach to a preceding noun phrase
The attachment of "today" in that sentence is syntactically ambiguous, just as the attachment of a prepositional phrase like "on this day" would be syntactically ambiguous.
My first interpretation was the same as lly's: I felt like "today" modifies "condition" instead of "described". I suppose "today" could modify the clause, as jlovegren suggests, but I do not think that is the only possibility.
Here is an example of a sentence where I think it's clear that today is not a clause-level adjunct:
"I mean that Mr. Vane's son is responsible for his condition to-day."
(Mr. Crewe's Career, by Winston Churchill (not the politician), 1908)
This doesn't mean the same thing as "Today, Mr. Vane's son is responsible for his condition." In the quoted sentence from Churchill, the word today is not used to locate the time when the son is responsible; it is used to locate the time of Mr. Vane's condition. In other words, the meaning of the sentence is closer to "Mr. Vane's son is responsible for his current condition" than to "Mr. Vane's son is currently responsible for his condition."
Another example where today is not a clause-level modifier:
So, in the future, our normal condition today will be considered disabled.
(The Realities of 'Reality' - Part II: Making Sense of Why Modern Science Advances (Volume 1), by Fritz Dufour, 2018)
Obviously "today" in this sentence could not be functioning as a clause-level adjunct, because the clause is talking about the distant future.
And a third example, this time in a clause with a past-tense verb:
The situation today was not predictable a generation or two ago, and the future too is largely unknown.
(Youth On Religion: The Development, Negotiation and Impact of Faith and Non-Faith Identity, by Nicola Madge, Peter Hemming, and Kevin Stenson, first published 2014, p. 217)
Parts of speech
I think it might not have been a good idea to combine this question with the part-of-speech question.
The part of speech of "today" seems complicated. My impression is that it functions like a noun phrase that functions like a prepositional phrase. (Hopefully that's not incomprehensible.) Or perhaps it could be said to function as a pronoun "fused" with a preposition. Prepositional phrases sometimes function like adverb(ial) phrases, but I don't know whether there's anything particularly adverbial about today in your sentence.
Etymologically, today obviously has the form of a prepositional phrase: the first part is from the preposition to, and the second part is from the noun day.
The word functions like a noun phrase in sentences like "today is a good day", and it clearly has a deictic meaning: these facts support the categorization of the word as a pronoun. ("Today" can also be used as just a regular noun, but I don't think that use is relevant.)
Like the noun phrases "that day" or "the day I went on my trip", "today" can be used adverbially ("I saw the city [that day/the day I went on my trip/today]". Noun phrases used this way have been described as "bare NP adverbs of time" ("Bare NP Adverbs", Richard K. Larson). It's clear that there is no adverb or (explicit) preposition in the phrase that day, so if the usage is truly analogous, I'm not sure whether even this usage of today warrants classifying the word as an adverb or preposition rather than as a pronoun.
In the sentence "Police described his condition today as serious but stable," it does seem to be possible to replace "today" with the NP "that day" and get another grammatical sentence: "Police described his condition that day as serious but stable".
So I think that today is a pronoun in that sentence, but I'm not sure how to describe the syntactic function. It might involve a syntactic structure that is also used when adverbs modify nouns.
Your previous question about home seems relevant, as it touches on similar issues.