You need a separate strand for each side of the neck, so that’s why they want you to join a new ball for that. Since all they’re having you do, though, is purl one more row, you could do that on the one side, then use that yarn to do the other side. Know what I mean?
I’m not familiar with the pattern itself, but it sounds like the first 10, 13, 10 stitches are the back. Knit the 10, put the center 13 on the holder and work the next 10–either with it’s own strand or after you finish the first 10.
Then the remaining 20 are used for the next portion of the directions.