This is actually a less straightforward question than it might seem, for a couple of reasons, although most of them come back to one source: transmission. No text - and here for simplicity’s sake we’ll assume a literary text, rather than a formal inscription or any of the various kinds of less formalized writing - from the ancient world that survives to today survives as a holograph (that is, as a text written in the author’s own hand). No version of Augustine’s Confessions, for example, was actually written down by Augustine in his own hand. The same is true of all the extant literary output from antiquity, from Republican authors like Ennius or Cicero to late Antiquity.

So when we want to know what a texts was made out of, we have to think about several different material texts: the surviving version (for literary texts, this is likely a later, medieval manuscript tradition, although some are preserved on papyri) which was copied later by third parties sometime after the date of composition, the several ‘formal’ drafts that were produced as part of a text’s circulation and distributed at/during the time of composition and editing among an author’s social circle, and, very likely, the most informal stages of textual composition, when an author actually drafted it.

The first of these is the easiest to find. Every critical edition of an ancient text will include a textual history (who has edited this text before? What manuscripts did they have available to them? Which did they use? What sort of edition was it?) and a stemma (which is a sort of manuscript family tree, showing how scholars think one manuscript is related to another and reconstructing what we know of each), and a number of translations and commentaries will have a somewhat simplified version of the same (you don’t usually need a list of every manuscript edition of a text in order to read a translation, but the translator might include a small paragraph giving you the general idea of what’s available and what kind of shape the text is in). Looking up the Confessions, the editio princeps (the first modern edition), to pick an example, was made in 1506 and was based on a trio of closely related manuscripts that were made in the ninth to eleventh centuries, but there are a number of other medieval codices used for more modern editions. Based on date alone, we can assume these codices are parchment. Since the manuscript tradition is how we have most complete literary texts, Augustine’s text is certainly not unique in this respect, and the editors of any text that is preserved solely on papyus will note that in the textual history.

But the Confessions wasn’t written in the ninth century CE, when the first of that family of manuscripts was created. It was written around the year 400 CE. So at this point, it’s not unreasonable to ask: okay, so what was Augustine’s text of the Confessions written on? There are complications here too, though. Brushing over for the meantime the complication that is the public circulation of a text in antiquity, here we’re left without a definite answer. For a Republican author, we can safely assume that a text was finalized and circulated as a papyrus roll. But by Augustine’s time, things have changed, and we’re stuck weighing probabilities somewhat. By this point, the codex has started to supplant the roll form as the medium for high quality texts, and this shift frequently comes with an accompanying shift toward parchment over papyrus. The Confessions in particular was almost certainly a codex. There is a statistical correlation between content and form here: Christian texts (like the Confessions) were more likely than others to be early codex-adopters. The Confessions references books within the narrative that are clearly codices rather than rolls (the famous tolle, lege scene must be a codex rather than a roll, but this is, again, to be expected of a Christian text), but to the best of my memory does not explicitly refer to its own textuality at any point (I’ll concede that it’s very possible I’m missing a reference within that specific narrative, but since we’re just using that text as an example the point stands), so while we can say that, as a Christian text of that period it was very, very, very likely a codex, we have to leave some room for doubt here. Plus this is a general trend, and Augustine is writing in the middle of a sea change. So while Augustine's Christian works probably circulated as codices we can't be so sure for every author even writing at the exact same time, and honestly we can't even say the same for his earlier, non-Christian works. And, furthermore, the shift to codex form is not exactly equivalent with a shift toward parchment over papyrus. Papyrus continues to be used for codices even after this shift occurs, and even if the Confessions first circulated as a codex we can’t say for sure whether its original form, such as an original form existed at all, was parchment or papyrus.

As an end note, I should also point out that it’s very likely that the Confessions was drafted in an entirely separate form: the tablet. Tablets weren’t thought to be suitable for high quality finished texts and would have been completely impractical for circulating the full text of something as long as (or even much shorter than) the Confessions, but they were thought to be useful for drafting work, since they allowed for much easier erasures and corrections.

So medieval manuscript editions of the text were certainly parchment, contemporary versions of Christian texts were almost certainly codex form but could have been either parchment or papyrus, and informal composition work likely started out on a tablet before ending up anywhere near either parchment or papyrus.

Moving your time frame a little ahead or a little back makes an enormous difference on whether an ancient text was more likely to be roll or codex, though, and cultural standards and frameworks matter quite a lot here, so this isn't necessarily the same throughout the ancient world.