Here the printer’s manuscript reads “a part … the part”. The 1830 typesetter couldn’t make much sense of this, so he replaced the definite article the with the indefinite article a, thus producing “a part ...a part”. There is another example of “a part ...a part” in the Book of Mormon text:
As David Calabro points out (personal communication), here in Jacob 5:45 an original and a could have been misinterpreted as and the during the early transmission of the text. If such an error occurred, it would have probably occurred as Joseph Smith dictated the text to Oliver Cowdery, especially if the d of the and had been pronounced by Joseph (compare and a /ændß/ with and the /ænddß/).
A completely parallel example in the olive tree allegory itself suggests that the original reading in Jacob 5:45 was actually “a part … the other part”:
In other words, while copying, Oliver Cowdery dropped out the word other, thus producing the incomprehensible “the part”. Under this analysis the occurrence of the definite article the in the printer’s manuscript is itself correct and is not an error for a.
Further support for the definite article the can be found later on in the book of Alma, where the original text apparently read “a part … the remainder part”:
This reading is based on the printer’s manuscript; the reading in the original manuscript is only partially extant, but the definite article the is clearly in 𝓞.
Theoretically, either “a part … the other part” or “a part … a part” will work as an appropriate emendation in Jacob 5:45. But the parallel use of “a part … the other part” in verse 25 of the allegory supports emending the text by inserting the word other in verse 45.
Summary: Replace “a part … a part” in Jacob 5:45 with the parallel expression “a part … the other part”, as implied by “a part … the other part” in Jacob 5:25 and by the anomalous reading of the printer’s manuscript in Jacob 5:45 itself (“a part … the part”), with its use of the definite article the.