In 2 Timothy 4:13, Paul asks Timothy to bring the scrolls & the parchments (When you come, bring the cloak that I left with Carpus at Troas, also the books [““Scholars widely regard τὰ βιβλία — the books — as a reference to books of the Old Testament, most likely on scrolls.” (Kruger, Canon Revisited); see, e.g., Luke 4:17, “And the scroll [Βιβλιον] of the prophet Isaiah was given to him. He unrolled the scroll and found the place where it was written”]. Thus, while the reference to the Old Testament Scripture is plain, what should be understood by “parchments”?
William D. Mounce, in the Word Bible Commentary, Pastoral Epistles, explains:
μεμβράνα is a Latin loan word for “parchment,” a writing material more expensive than papyrus, capable of being reused and more durable, made from the skins of sheep and goats. Kelly (216) argues that the word was commonly used of a codex (as opposed to a scroll). μάλιστα can mean “especially” (cf. discussion in Comment on 1 Tim 4:10), in which case the parchments are in addition to the books. It can also be an identifier, “that is, namely, to be precise,” in which case the books are more closely defined as the parchments (Skeat, JTS n.s. 30  173–77). Only Paul, Carpus, and perhaps Timothy knew what they contained.
The New American Bible Commentary (Lea) tentatively follows the suggestion that the scrolls & parchments refers to the same thing:
T. C. Skeat has suggested a view of the latter phrase of v. 13 which links the scrolls and the parchments together. Considering it unlikely that Paul would carry a library with him, Skeat views the adverb “especially” (malista) as equating the “scrolls” and the “parchments” instead of differentiating between them. In his view Paul would have been saying, “Bring the books—I mean the parchment notebooks.”This view still leaves us uncertain about the contents of the books, but Skeat’s explanation seems the best solution.
George W. Knight III in the New International Commentary on the Greek Text also favors Skeat.
Micheal Kruger offers the plausible explanation that the parchments were actually copies of Paul’s own letters:
As for the content of these codices (or notebooks), a number of suggestions have been made over the years. Given that Paul distinguishes these codices from the Old Testament writings, many scholars have argued that they likely contained some sort of Christian writings. This may have included a variety of things, such as excerpts of Jesus’s teachings or early Christian testimonia (Old Testament proof texts supporting Messianic claims about Jesus). However, one of the most compelling possibilities is that these notebooks contained (among other things) copies of Paul’s own letters. It was not at all unusual in the Greco-Roman world to keep copies of (and even publish) one’s own letters. Cicero exemplifies this practice as his personal secretary, Tiro, kept extensive copies of his letters.Cicero would occasionally receive a complaint from friends that one of their letters (from Cicero) was lost or damaged; on such occasions Cicero would quickly dispatch a replacement copy from his own collection. And where did Cicero make and keep copies of his letters? He tells us: “I am jotting down a copy of this letter into my notebook.”In other words, Cicero kept copies of his[…]
Excerpt From: Michael J. Kruger. Canon Revisited: Establishing the Origins and Authority of the New Testament Books. Crossway, 2012. iBooks. Kruger’s explanation — more developed than most explanations is not an untenable or unparalleled suggestion. John MacArthur also permits this is an explanation:
These particular parchments may have contained copies of Paul’s own letters or may have been blank sheets on which he planned to write other letters. He had no plans to finish studying or to finish writing.
Ronald Black and Ronald McClung also offers that the parchments may have contained Paul’s own correspondence:
Whatever their form, Paul was asking for his books. Almost certainly, copies of the Scripture were among them, along with other works. Paul’s own notebooks and copies of his correspondence would likely have been among the scrolls, too, and it is possible that legal documents, such as proof of his Roman citizenship, were included as well.
Aside: Michael Kruger notes Skeat’s position in footnote 99 of chapter 7:
“T. C. Skeat, “‘Especially the Parchments’: A Note on 2 Timothy iv.13,” JTS 30 (1979): 173–77, has argued that these two kinds of writings are one and the same. He understands Paul to be saying, “Bring the books, that is (μάλιστα) the parchments.” However, this suggestion has gained only limited support. See discussion in Stanton, “Why Were Early Christians Addicted to the Codex?,” 177–78”