In brief: only some dispensationalists believe that Jews and Gentiles are currently under two completely separate covenants – most instead believe that there is one New Covenant that is applied differently to Jews and Gentiles.
However, either way, the "old covenant" of Hebrews 8:13 is seen to be the Mosaic Covenant, as distinguished from the Abrahamic Covenant (which the New Covenant is based on).
Now for the details and some sources. Dispensationalists like the authors of the Moody Bible Commentary and Thomas Constable understand the "obsolete" covenant of Hebrews 8:13 to be the Mosaic Covenant. This is indicated by the references to the priestly and sacrificial system in Hebrews 8:1–5 which does literally "disappear" in the fall of Jerusalem in AD 70.
The Mosaic Covenant (obsolete, disappearing) is thus distinguished from the Abrahamic Covenant, as Thomas Constable writes:
The New Covenant is a branch of the Abrahamic Covenant. In the Abrahamic Covenant, God promised Abraham a piece of real estate for his descendants, an incalculable number of descendants, and blessing for his descendants and for all people through his descendants (Gen. 12:1-7; et al.). [...] The Davidic Covenant gave more information about God's promises regarding descendants (2 Sam. 7). The New Covenant revealed the particulars of the promised blessing (Jer. 31).
Each of these later covenants relates to the Abrahamic Covenant organically; they were outgrowths of it. In contrast, the Mosaic (Old) Covenant does not relate organically but "was added" (Gal. 3:19), as an appendage. It explained how the Israelites could maximize the benefits God had promised in the Abrahamic Covenant. Consequently, when God terminated the Old Covenant, it did not eliminate anything He had promised Abraham. (source; bold added)
Thus the Abrahamic Covenant reaches its fulfillment in the New Covenant. Most dispensationalists, as Constable writes, believe that there is only one New Covenant, though it is applied differently to Jews and to Gentiles. But he does note that a few dispensationalists believe that there are two new covenants, one for Jews and one for Gentiles:
Dispensational commentators have taken various positions on the relationship of the New Covenant, promised in Jeremiah 31, to the New Testament references to the New Covenant. Was it the same covenant, or is a second New Covenant in view? Some believe there are two new covenants: one with Israel and one with the church. This position rests on the fact that the New Covenant, promised in Jeremiah 31, was specifically "with the house of Israel and with the house of Judah" (Jer. 31:31). Those who hold this view take the New Covenant, under which Christians live, as a different "New Covenant" (2 Cor. 3:6; Heb. 8:8; 9:15). They regard Jesus' references to the "New Covenant" as to a New Covenant with the church (Luke 22:20; cf. 1 Cor. 11:25). (source)
This is not the view that Constable prefers, however, and he indicates that "few writers today" take this position.